I have a case on my square-1 that I can't solve where only two edges of one layer are swapped. I am not looking for an algorithm as much as for an explenation how that is possible. I can 3-cycle edges pure and also corners can be 3-cycled just like on a 3x3x2. Since edges and corners behave exactly the same on the puzzle (8 "identical" pieces) how can such a two-swap parity occur on only one part of pieces? If I do a 90° turn I get rid of the edge parity but create a corner parity.
What's happening there?
As you have noticed, when the puzzle is in the cubic shape, a dihedral turn swaps two corners and two edges and a 2 swaps for the corners in even and 2 swaps for the edges is even. As long as you stay in a cubic state you can't find a move that violates this.
When you move out of a cubic state though you can arrange different number of swaps for the edges and corners. For example, what Jaap calls the "Scallop" pattern
If you arrange both the top and bottom into this pattern then if you do a dihedral turn with 3 corners swapping with 3 other corners you have changed the parity of the corners independent of the edges. This is just one example of many. I haven't reverse engineered any "parity fix" routines to see what swaps there are doing but i suspect they aren't using the Scallop pattern.