Singmaster ("Notes on the 'Magic Cube'", p. 12 of edition 4) gives 12 classes or
"orbits" of assemblies, each one such that the other 11 cannot be reached without
disassembling the cube. To quote Singmaster, "This also means that if we
reassemble the cube at random, there is only a 1/12 chance of being able to get
back to START," and, "It is advisable to reassemble in the starting pattern."
Singmaster's notes have several pages of relevant group theory for those