From there the solution should be straightforward (e.g. two intersecting
3-cycles).

Finally, it seems clear that this entire problem --and all the subsequent
discussion-- maps directly onto a virtually identical problem on the 4by
cube (i.e. simply be removing the center planes).

>Note that you get an apparant parity reversal by flipping the cubies, but
>this does not actually move anything.  In other words, no amount of
>flipping and moving will allow you to end up moving A->B->C->A.  That's
>why I solve edges first.

Again, perhaps I'm missing the point, but if you don't care about how
the flipping comes out, the A->B->C->A  3-cycle is certainly doable:

For example:

[WARNING: EVEN MORE BORING STUFF AHEAD!!   ;]

(I have no idea how to show this notationally, so I'll try pictorially.)

                         |
1]                    2] V                  3]
 X|A|X|C|X             X|Y|X|C|X           ->Z|Z|Z|Z|Z
 X|X|X|X|X             X|Y|X|X|X             X|Y|X|X|X
 X|X|X|X|X             X|Y|X|X|X             X|Y|X|X|X
 X|X|X|X|X             X|Y|X|X|X             X|Y|X|X|X
 X|X|X|B|X             X|A|X|B|X             X|A|X|B|X


4]                   5]                     6]
 Z|A|Z|Z|Z             X|B|X|X|X             X|B|X|X|X
 X|X|X|X|X             X|X|X|X|X             X|X|X|X|X
 X|X|X|X|X             X|X|X|X|X             X|X|X|X|X
 X|X|X|X|X             X|X|X|X|X             X|X|X|X|X
 X|X|X|B|X             Z|Z|Z|A|Z           ->?|?|?|?|?
   ^
   |                  [Rotate Face           
                       one-half turn]

      >---
7]        \           8]                    9]
 X|B|X|X|X \           X|B|X|X|X             Z|C|Z|Z|Z
 X|X|X|X|X |           X|X|X|X|X             X|X|X|X|X
 X|X|X|X|X |           X|X|X|X|X             X|X|X|X|X
 X|X|X|X|X V           X|X|X|X|X             X|X|X|X|X
 ?|?|?|?|?             Z|Z|Z|C|Z<-           X|X|X|B|X 
      
 [Rotate next-                              [Rotate Face
  to-bottom                                  one-half turn]
  plane 1/4 Turn]

                            <----
10]                   11]        \          12]
  Z|C|Z|Z|Z             Z|C|Z|Z|Z \           Z|C|Z|Z|Z
  X|X|X|X|X             X|X|X|X|X |           X|X|X|X|X
  X|X|X|X|X             X|X|X|X|X |           X|X|X|X|X
  X|X|X|X|X             X|X|X|X|X ^           X|X|X|X|X
->?|?|?|?|?             ?|?|?|?|?             X|X|X|A|X<-

                        [Rotate next-
                         to-bottom
                         plane 1/4 turn]

    |
13] V                 14]                   15]
  Z|Z|Z|Z|Z             X|Y|X|B|X<-           X|C|X|B|X
  X|Y|X|X|X             X|Y|X|X|X             X|X|X|X|X
  X|Y|X|X|X             X|Y|X|X|X             X|X|X|X|X
  X|Y|X|X|X             X|Y|X|X|X             X|X|X|X|X
  X|C|X|A|X             X|C|X|A|X             X|X|X|A|X
                                                ^
                                                |


                         Cub.E.D