If you want to see this bad boy in all its gear turning glory watch this.

If the edges around the vertical middle layer need to twist like this blue purple one does, do 4L'. L' means L up and 4L' means 4 180˚ turns of the L layer going up. On a normal cube a 180˚ turn has the same affect whether you go up or down, but because the middle layer of the Gear Cube turns with the outer layer, direction matters.

If the edges around the vertical middle layer need to twist like this Orange Purple one does, do 4R.

This is the result of the above move. Notice that none of the other
pieces of the cube were affected. Only the 4 edge pieces around the
vertical middle slice twisted into place.

To get from solved to here, or from here to solved, do R 6U R'. It swaps the top and bottom centers and it swaps the right middle layer row with the left middle layer row.

To get from here to solved you can do R 2U' R'. It swaps the top and bottom centers and it swaps the right middle layer
row with the left middle layer row. And it makes the tweaked edges flat. If the blue was up on the tweaked edge in the picture instead of red, then do R 2U R' to solve it.

Or you could do R 6U R' followed by the rotate edges algorithm. This is longer, for sure, but you don't need to remember a separate algorithm.

To get from solved to here do (R U R') D (R U' R') D'. It does 3 3-cycles. Centers: F to U to R, and B to D to L. And it cycles 3 Y-shaped sections. The ULF to DRF to DLF. To get from here to solved do the opposite moves in the opposite order. That is, D R U R' D' R U' R'.

Here is a video explaining the 3-cycles described above.

Up Replace Down 3-Cycle Gear Cube Style

For the following table start from solved with red in front, yellow on top, and green on the right. Purple on the Gear Cube is opposite red and orange is opposite yellow, so I'm pretending my "normal" cube is colored that way. M is the layer between R and L. The results reported below are on the top of the cube after the move. * The edges can't be tweaked on a normal cube.