Rival types are where you take the ending letter and flip it. You might be thinking "It's only a one-letter difference, how does that make them rivals?" and I am here to answer that question. That one letter actually changes all of the cognitive functions. Let's take an Isfp and Isfj for example.
Isfp functions
Driver: (Fi) Introverted Feeling (Authenticity)
Co-Pilot: (Se) Extroverted Sensing (Sensation)
10yr old: (Ni) Introverted Intuition (Perspectives)
3yr old: (Te) Extroverted Thinking (Efficiency)
Isfj functions
Driver: (Si) Introverted Sensing (Memory)
Co-Pilot: (Fe) Extroverted Feeling (Harmony)
10yr old: (Ti) Introverted Thinking (Accuracy)
3yr old: (Ne) Extroverted Intuition (Exploration)
Now, if you read the most different types section you might be a bit confused. This one also doesn't share any of the same cognitive functions or order of the functions. I will expand a bit on most different types basically, those types are even more opposite than this because the two dominant functions of one type would be inferior functions of their most different types and also the introvert/extrovert flip of the function (I hate how many times I've said the same word). Anyways, as you might have observed with this example they have sensing and feeling both as dominant functions just in different places. Continuing on with the explanation, the reason they are rivals is that they have similar ending points that they heading toward but are going about it very differently. So let's say there was a fork in the road to their destination and the Isfj goes left while the Isfp goes right. The Isfj starts asking why Isfp is going the wrong way and the Isfp asks the Isfj the same thing. Little do either one of them know that they both have different directions on their GPS but it will bring them to the same place. Both are annoyed at each other because of the fact that they believe the other is doing it "Wrong." I hope that annalogy helped.