I take back my statement about solving the cube, also: I do have one transform that is the product of an order-3 centers-only cycle and an order-5 edges-only cycle, and I can use that transform to do very nearly everything...except for one maneuver. If the normal layout of the edges on one face is: a b h c g d f e Then, when I try to solve the cube, once in a while I end up with one of the following two equivalent forms: ~a ~b c b h c or h a g d g d f e f e I have not yet fully determined exactly which of the two identical edges of each color is where. In particular, I strongly suspect that the first form is really `~b ~a' (that is, if you think of the two edges together as a single C^3 edge, then that edge was flipped), while I suspect that the second form is correctly rendered.
I have found that if I do sensible things to the cube, I can mostly work until
the cows come home and not fix the thing, but if I screw up a transform and
then have to recover from a somewhat scrambled mess, more often than not it
comes out OK when I get the cube back in shape. sigh. right now I'm working
on a group-theoretic analysis to try to get some idea for what type of move or
conjugate I have to be looking for in order to make the thing work out.