Thanks to all who responded. I haven't yet got what I consider a
solution for my problem (shift a slice and resolve is my current
method which is slow and ugly) but at least I understand my problem
A few questions:
1. What is the definition of parity by which commutators are even,
but slice turns are odd? I haven't been able to come up with a
cube-wide parity. (I know no group theory).
2. How many orbits does the order 4 cube have? I can only think of
three (twirling a corner cubie). Then again, I haven't painted the
facelets yet, so there could be orbits I haven't begun to see
3. Would an order 6 cube have any challenge beyond the order 4? I
think the answer is no--if you are able to solve the 3-cube and the
4-cube you can solve any cube.