[avpsclrh] Scrambling center squares of a Rubik's cube

In a solved 3x3 Rubik's cube, if each center square could be independently rotated 4 different ways, then there would be 4^6=4096 visually identical solved states.  However, only half of them, 2048, are possible.  This can be computed by comparing the number of possible states of a supercube versus regular cube.

For 4x4, if everything were independent, the 4 center squares of 1 face could be permuted and rotated in 6144 = 2^11 * 3 different ways, yielding a hypothetical total of 6144^6 ~= 5.4e22.  However, the actual total number is 95551488 = 2^17 * 3^6 = (2^3 * 3)^6/2, so less by a factor of 2^49.

Which center face states are possible, and what are some algorithms to reach them?