Just a clarification (and a discussion promoter, perhaps). I can
believe it is possible to put the center 2x2 blocks of a 4x4x4
cube into any relative position. But, there is still only one
way to solve it. This is because the relative position in the
solved state is determined by the corners. I guess this allows
for a new class of pretty patterns, especially DOTS:
DOTS as we know it in the 3x3x3 (two sets of three; one
set clockwise, the other couterclockwise)
DOTS (two sets of three; both (counter)clockwise)
DOTS (two sets of three; similar to BASEBALL)
DOTS (three sets of two; like Christman's Cross)
DOTS (three sets of two; pairs opposite) (a local max)
DOTS (three sets of two; all pairs adjacent)
DOTS (one set of six; similar to the checkboard obtained
by Plummer's Cross + dots + pons)
etc.

Anybody want to take a shot at trying to catalog all the local maxima?