[vlswaxin] Continuous twisty

Cut a sphere along any plane, rotate the pieces by any angle, and glue it back together to a sphere.  Repeat.

This is a continuous generalization of a Rubik's cube, probably realizable only virtually, and probably too difficult as a practical puzzle.

We can apply the same idea to a cube, maintaining the constraint that, after a twist, it must remain a cube.  I think only 60, 90, and 180 twists are possible along certain families of planes.

Although there are a few special planes in which the cross section is a regular hexagon, I don't think a 30-degree twist results in a cube.  What shape results?

We could drop the constraint that the shape remain constant, allow arbitrary rotations and translations, likely resulting in wild, highly disconnected, non-convex shapes.

Are these puzzles one-way functions suitable for cryptography? Or will it be too obvious that generally only one reversal step is possible from a given state?

Inspired by the Lament Configuration.