My own heterodox chess
If I were a chess variant enthusiast who invents and proposes his own variants, I might propose some of the following. The nicest are maybe §2 (with crowned bishops), §3, and §6. -- As far as already existing variants are concerned, my favourite ones are listed in the homepage of my chess web space.
§1 Paradigm dragon Seirawan
Seirawan Chess, with the following change: the special pieces are a bishop-xiangqihorse hybrid (aka Paradigm dragon) and a rook-xiangqihorse hybrid. (One can propose an equivalent change for Illescas Dragon Chess as well, or for any kind of Capablanca/Gothic Chess. For other similar variants, where the knight-queen hybrid is present too, this piece can be replaced by a queen-xiangqihorse hybrid. -- You can try some sort of Paradigm dragon Seirawan, as a custom variant, on www.chess.com/variants/custom: with two Paradigm dragons, since the other hybrid is not available.)
§2 Modified versions of Paradigm Chess30
I love Paradigm Chess30. Nevertheless, sometimes I think those modified bishops are too strong. Is that fair? Can rooks still be called major pieces, in Paradigm Chess30? And, poor knights... On the other hand, the fans of Seirawan Chess or Capablanca/Gothic Chess will find that Paradigm dragons are not powerful enough! For all these reasons, here's a list of suggestions for modified versions of Paradigm.
[Replace the rooks too.] Standard chess, with the following change: each player has bishop-xiangqihorse hybrids (aka Paradigm dragons) and rook-xiangqihorse hybrids, in the place of bishops and rooks, respectively. Castling as in standard chess. Pawns can promote to any of the pieces that are present in this variant: no rooks, no bishops. This variant can be seen as a modified Paradigm Chess30, and can be played with the same type(s) of randomization -- either 30 or 1230, see section §5 of my web page on Paradigm Chess30. (By the way, I think that standard chess starting position is always a valid alternative, even for Paradigm Chess and for all these modified versions.)
[Replace the rooks and the queen too.] The same as above, but the queen is replaced too: here's a queen-xiangqihorse hybrid in the place of the queen. Pawn promotion, same principle as above. Randomization(s) as above.
[Very powerful piece replacements: leaping hybrids.] Paradigm Chess, or any of the above Paradigm subvariants, with knight hybrids precisely in the place of xiangqi horse hybrids: so, knight-bishop hybrids instead of Paradigm dragons, etc. Pawn promotion, same principle as above. Randomization(s) as above.
[Put away the dragon wings and crown your bishops.] Standard chess, with the following change: each player has non-royal king-bishop hybrids (aka wazir-bishop hybrids or crowned bishops or missionaries, in fairy chess terms) in the place of the bishops. Pawn promotion, same principle as above. Randomization(s) as above. -- Further subvariants: you could crown the rooks too (and/or the knights, for which see the next point).
[Replace the knights!] Paradigm Chess, or any of the above subvariants, with modified knights: (a) non-royal king-knight hybrids (aka crowned knights) in the place of the knights, or (b) non-royal king-xiangqihorse hybrids (aka crowned xiangqi horses) in the place of the knights. Pawn promotion, same principle as above. Randomization(s) as above. -- This simple idea, crowning the knights/horses (or the rooks, or both!), can be applied to standard chess too; no Paradigm-style randomization in this case (but 18 and 960 are possible, of course).
[Peaceful crowning subvariants.] For each of the above subvariants involving non-royal king hybrids (aka crowned pieces), we can propose a further subvariant with peaceful crowning: the king move of peaceful-crowned pieces cannot be a capture -- not even a potential king-capturing move, for check and checkmate. (The diagonal moves of crowned bishops never count as king moves; the same is valid for the orthogonal moves of crowned rooks. -- My idea of peaceful crowning, probably not new, was inspired by the rules of the Spartan Chess piece called lieutenant or tagmatarchos.)
[Peaceful knighting/horsing too?] Peaceful knighting follows the same principle as peaceful crowning: the knight move of peaceful-knighted pieces cannot be a capture -- not even a potential king-capturing move, for check and checkmate. The same can be proposed for xiangqi horse hybrids too. For example, a bishop-xiangqihorse hybrid with peaceful horsing will be a very interesting weakened version of a Paradigm dragon: one should try a Paradigm Chess subvariant with this change. The principle of peaceful knighting/horsing can be applied to any (sub)variants where knight hybrids or xiangqi horse hybrids are present -- consider peaceful-knighted or peaceful-horsed kings too, either royal or non-royal!
§3 Turn pawns into xiangqi horses, or into other minor pieces, with no need of reaching the eighth rank
Standard chess, adding the following rule: the pawn-to-xiangqihorse change. Pawns follow the standard rules, including the two-square advance, en passant, and promotion. But, in addition to that, each pawn can turn into a xiangqi horse, according to either pattern (1) or pattern (2). (1) Make a legal non-promoting pawn move, capturing or not, and then change the same pawn to a xiangqi horse, in the same turn; (2) change a pawn to a xiangqi horse, and then make a legal move with that same horse, capturing or not, in the same turn. The pawn-to-xiangqihorse change is always irreversible. The check and checkmate rules remain unchanged: you can give check or checkmate with this special move and, on the other hand, it can be a get-out-of-check move. (The first half of the special move is not relevant or valid for check and checkmate.) Promotion remains unchanged too: no promotion to xiangqi horse (if it were allowed it would be a senseless choice, since a horse is even weaker than knights and bishops). Two further subvariants are possibile, with fixed order: move first and then change to xiangqi horse; or, inversely, change to xiangqi horse first and then move. Move-and-then-change is similar to promotion: therefore, it may be considered more elegant and chess-like, and that's one of the reasons why I think this fixed-order subvariant should be the default choice. Note that, if change-and-then-move is allowed, one must pay attention to heterodox (distance) pawn attacks, including checks and checkmates!
The same as above, but each pawn-to-xiangqihorse change takes one full turn, with no movement or capture. You are always allowed to make this special move, for any of the pawns you have -- unless you are in check, of course. (This subvariant is simpler, but probably less interesting.)
The same as above, in any subvariant (free order / move and then change / change and then move / change only), but pawns may turn into different minor pieces: (a) xiangqi horses only, as already seen; (b) knights; (c) bishops; (d) knights and bishops; (e) non-royal kings; (f) non-royal kings and knights; (g) non-royal kings and bishops; (h) non-royal kings and knights and bishops. In all cases, I think promotion would better remain unchanged: pawns promote to standard chess non-royal pieces only (most likely a queen, of course).
Set a limit! How many times can each player make the special move in a game? The starting idea was setting no limit (i.e. eight times, since the pawns are eight), but it's probably more sensible to establish a maximum such as three, two, or only one, also depending on the individual subvariants. So, choose any number, from one to eight.
§4 Variants with a more powerful king
Standard chess, with the following change: each player has a royal king-xiangqihorse hybrid in the place of the king.
Standard chess, with the following change: each player has a royal king-knight hybrid in the place of the king. (I think this variant already exists. For sure it can be set as a custom variant on www.chess.com/variants/custom. I have played several games and like it.)
§5 Other xiangqi-inspired and Synochess-inspired variants: cannon or non-royal king instead of queen
Standard chess, with the following change: each player has a cannon in the place of the queen. (The same change applies to pawn promotion: no queen is allowed -- you can promote to a cannon, but a rook will usually be better of course.)
Exactly the same as above, with a non-royal king in the place of the queen. (I suppose this variant may already exist. I have tried it, as a custom variant, on www.chess.com/variants/custom, but I can't bear the Chess Dot Com symbol convention for royal vs. non-royal kings, when both are present.)
§6 A multifaceted variant with weakened pieces, inspired by ChinKor chess (that is xiangqi and janggi)
Standard chess, with at least the following changes: xiangqi horses (aka ChinKor horses) replace the knights, and ChinKor elephants replace the bishops. There is no palace and no river: all pieces can move throughout the board.
Elephant subvariants. The elephants can be: (a) Chinese elephants; or (b) Korean elephants; or (c) Chinese-plus-Korean elephants, meaning that both the xiangqi move and the janggi move are allowed; (d) non-leaping ferz-alfil hybrids, aka non-leaping ferfils/falafels (1-or-2-square diagonal move); (e) leaping Chinese elephants, aka alfils/alpils/pils; (f) leaping Korean elephants; (g) leaping Chinese-plus-Korean elephants; (h) leaping counterpart of (d), that is ferz-alfil hybrids, aka ferfils/falafels or modern elephants or some further silly and useless names. The default choice should be (d), because that is a reasonably weakened (short-range) bishop, and is also very Chinese-flavoured, since it can be seen as a perfect advisor-elephant hybrid, in xiangqi terms; but (c-g-h) are very stimulating possibilities too. Hybrid versions of Korean elephants could be proposed as well.
Queen subvariants. (a) A cannon replaces the queen (cfr. section §5); or (b) a non-royal king replaces the queen, as in Synochess (cfr. section §5 again); or (c) a weakened short-range queen replaces the queen (1-or-2-square move: a non-leaping ferz-alfil-wazir-dabbaba hybrid in fairy chess terms -- also called a bodyguard!); or (d) leaping counterpart of (c); or (e) a rook-xiangqielephant hybrid replaces the queen; or (f) a rook-janggielephant hybrid replaces the queen; or (g) a rook-elephant hybrid replaces the queen, based on the very same elephant type that has been chosen as a bishop replacement (see Elephant subvariants above); alternatively, (h) the queen remains as in standard chess, which is the least ChinKor-like option. The default is (c), but cannon enthusiasts don't have to be sad! See Rook subvariants and Further cannon subvariants and rules.
Rook subvariants. A rook is a rook, with no variation of its rules. But, in the default subvariant, (a) each player has only one rook, the other being replaced by a cannon. You may castle either with the rook or with the cannon, following the same rules as in standard chess, if castling is allowed (see Castling subvariants). There is a setup choice before the game starts, similarly to the horse-elephant switch in janggi. White decides the position of his/her own rook and cannon (squares a1, h1); after that, based on White's decision, Black does the same on his/her side. Alternatively, (b) each player has two rooks, in the usual corner positions.
Further cannon subvariants and rules. (1) If each player has two rooks (see Rook subvariants above) and the pawns start on ranks 3 and 6 (see Pawn subvariants below), a white cannon and a black cannon will be inserted on ranks 2 and 7 respectively. In this case, too, there is a setup choice before the game starts: White places the cannon on one of the eight empty squares and, after that, Black decides the position of the cannon on his/her own side. (2) Xiangqi or janggi cannons? (a) The Chinese-rule cannon is default in this chess variant, but (b) the Korean-rule one is allowed too: see the general notes towards the end of this web page. (3) In any subvariant where cannons are present, either of the players cannot move his/her cannons in the first turn.
King subvariants. The face-off or facing-each-other rule: (a) does not exist; or (b) is the same as in xiangqi, that is vertical only; or (c) is the same as in Synochess, that is on files and ranks too; or (d) is queen-style, that is even on diagonals. The default is (b). In addition, the Korean variant of the rule could also be taken into consideration. (Non-royal kings are never affected by these rules. Note that the face-off rule regards the usual restrictions on castling too, if your subvariant allows castling.)
Castling subvariants. (a) The same as in standard chess; or (b) opposite castling, meaning that same-side castling is forbidden; or (c) no castling at all, which is the default.
Pawn subvariants. (a) The same as in standard chess, which is the default; or (b) standard chess pawns, but no two-square advance, and consequently no en passant; or (c) the same as (b) but with a change in the initial setup: the pawns start on ranks 3 and 6.
Promotion subvariants. (a) Pawns promote to non-royal kings only (default choice -- see also section §7); alternatively, (b) pawns promote to the pieces that are present in the initial setup.
Short suggestions for a basic subvariant. Standard chess, with the following changes: xiangqi horses replace the knights; short-range bishops replace the bishops; a short-range queen replaces the queen; pawns promote to non-royal kings only. (Short-range means: non-leaping, 1-or-2-square move.)
§7 Some weakened-promotion variants, approximately arranged from highest to lowest weakness
Standard chess, with the following change: pawns promote only to xiangqi horses.
Standard chess, with the following change: pawns promote only to bishops and knights. (I suppose this variant may already exist.)
Standard chess, with the following change: pawns promote only to cannons.
Standard chess, with the following change: pawns promote only to bishops, knights, and cannons.
Standard chess, with the following change: pawns promote only to non-royal kings. (I suppose this variant may already exist.)
Standard chess, with the following change: pawns promote only to non-royal king-xiangqihorse hybrids.
Standard chess, with the following change: pawns promote only to non-royal king-knight hybrids.
§8 See my page on Spartan Chess and other variants with more than one royal king per player (to do)
Game balance
I think that some subvariant combinations, especially in section §6, may perhaps give rise to balance problems. Some of my default choices are possibly questionable. In any case, standard chess and a lot of other important games are unbalanced too. Inverted-colours rematch is often a great solution, as is well known for Fischer Random starting positions, etc.
Xiangqi or janggi cannons?
A general note on the variants with cannons: we can choose either the Chinese-rule cannon or the Korean-rule one. They are both nice and interesting! (The choice must be stated in advance, of course, as a part of the ruleset. I don't know if having both types of cannon in any one individual variant could be interesting.)
KOTH and campmate subvariants
I like King of the Hill, and Couch Tomato's campmate rule as well. Either hillmate or campmate (not both together!) can be added to plenty of chess variants, including the ones listed in this web page.
Si parva licet componere magnis: other famous variants, put together with the ones of my list
A lot of existing and more or less important variants could combine with all or most variants listed in this web page: Chess 960, Duck Chess (!), Three-Check, Crazyhouse and Bughouse (!); Seirawan Chess and Illescas Dragon Chess (perhaps with some adjustments, see also section §1); Torpedo, Sideways Pawns, Capture Anything, etc.