[pkedeqfd] Flipping a face

Start with an NxN grid of squares, and select one row.  One can either rotate the row upside down (staying in the plane) or mirror reverse it the long way (flipping outside of the plane).  Flipping the short way is also possible but seems less interesting: the squares don't move.

These two moves are possibilities for a 2 dimensional analogue of the Rubik's cube.  The latter flip is, I think, more common, but is weird because the flip requires another dimension.

Carry over these ideas back to 3 dimensions.  Remove 9 cubies in a face, turn the face around, and stick it back on.  Or, mirror reverse a face through the center.

Inspired by thinking about 4D analogues, where physically impossible 3D moves become possible.