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.

/Bernie