The concept of the Rubik's cube can easily be extended to other polyhedra, and many such examples have been built. Extend it to polyhedra with 180 degree dihedral angles, namely tilings. This is best done virtually, which is especially convenient since screens are also flat.
Infinite tiling would of course be absurd. Toroidal, spherical, or Klein bottle depending on how the edges match up. Hyperbolic possible.
A little UI task of how to depict a move as it happens: perhaps paint flows from region to region.
Face turning, vertex turning, edge turning. Edge is attractive because you don't have to specify the direction to turn; either way gets the same result.
Extend to honeycombs in 3D. One possible large physical implementation could be a jungle gym in which you need to climb around to manipulate the puzzle. The colors go only on the vertices or edges, not faces or cells.
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