Strange Cube Index
This section shows how one cube can be transformed into another by a single move, swapping half the tope face with half the bottom. The transformation sequence is started from the regular cube (170+170). Each entry shows a starting cube and the set of possible next cubes.
Only cubes that can actually be reached are shown in this list.
You could use this page to help you navigate back from a jumbled cube to a proper cubic one. Find your cube in the left column and look for one in the right column that has lower depth. Repeating this process will get you back to the two squares in at most 7 moves.
But it might leave the middle row in a mess, but this is easy to resolve.
Unfortunately this page only deals with the shapes, the real puzzle also involves getting the coloured edge pieces into the correct positions.
(c) John Whitehouse 2017