David,

Interesting thoughts.

It seems to me that the icosahedron is a more attractive physical target than the
dodecahedron. It has 12 shear planes. A move would be a 72o rotation of a
five-triangle subsection. My intuition is that it is buildable more or less along
the line of the Rubik cube, except that the fixed axes point at vertices instead of
center faces.

The dual of the 2x2x2 cube seems to be an octahedron with shear planes along its
edges.

More generally slice a sphere a buncha times. The cuts are shear planes. Don't
let it fall apart (there is the trick!). Now paint the surface pieces. Now twist to
mix.

The actual shape does not seem to matter much. It is the interaction of the shear
planes that makes the puzzle.

Happy sphereing,

Bill