for what it's worth, i'll make some conjectures about mark's questions.

A) What is the next most commutative element (the pancentre?)
after the 12-flip?

(presumably, start excluded as well)

i'll guess that these four conjugacy classes are tied for next.

corner cycle structure:  (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1-)
edge cycle structure:    (1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)

corner cycle structure:  (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1-)
edge cycle structure:    (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)

corner cycle structure:  (1+)(1-)(1-)(1-)(1-)(1-)(1-)(1-)
edge cycle structure:    (1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)

corner cycle structure:  (1+)(1-)(1-)(1-)(1-)(1-)(1-)(1-)
edge cycle structure:    (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)(1+)

B) What is the least commutative element (the anticentre?) of
the cube group?

i'll guess

corners:  (1)(7)     edges:    (1)(11)
corners:  (1+)(7-)   edges:    (1)(11)
corners:  (1-)(7+)   edges:    (1)(11)
corners:  (1)(7)     edges:    (1+)(11+)
corners:  (1+)(7-)   edges:    (1+)(11+)
corners:  (1-)(7+)   edges:    (1+)(11+)

each of these splits into two conjugacy classes. i think this is the
example bandelow gives in his book.

mike