I had an interesting thought for a new puzzle the other day. It looks mostly like a 2x2x2, but two of the axes are slightly skewed so that four of the corners are fat and four of them are thin. The third slice, which I think of as being 'horizontal', can always be turned along. The other two need to be lined up properly in order to turn, with one fat and one thin piece needing to be on each side to make a slice.
From the initial position, it's possible to slice along either of the two vertical slices, but then because two fats and two thins are next to each other, you can only rotate the top 180 degrees before doing another slice, which returns the puzzle to the original shape. The other thing you can do is turn the top slightly off from 90 degrees, and then turn one of the vertical slices, which leaves the puzzle in its original shape in an interesting way.
Making this puzzle is an interesting challenge. I think making a skewed version of a regular 2x2x2 mechanism mostly works, although some of the internal pieces need to be cut up into a few more pieces. I leave the details as an exercise to the reader