According to Singmaster, there are 12 disjoint orbits (his term) of cube
positions. The article I read doesn't give a derivation, but the basic
idea is clear. It should be possible to characterize the orbit of a given
position as a function of the transformations that would have had to
be done on each cubie ( as if it could be moved alone ) to bring it
to its proposed position. Given such a charaterization, one could
declare positions fed to :CUBE illegal without trying to solve the
problem.

This relates to my thoughts on the problem of inversion of an unknown
transformation. Suppose you are given a Cube that is N unit moves
from home, determine the correct sequence to get it there. I think
a representation of the Cube in terms of combined tranformations on
the cubies is the place to start.
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