Yea in a 3x3x3 cube having the centers fixed relative to one another is really useful (and although i never really thought about it permutation limiting). I just moved to a 4x4x4 cube, which has non-fixed centers and for a while i was really lost. In fact in order for me to solve the 4x4x4 I HAVE to solve the centers first, i cant get my head around it any other way.
maybe this is a little off topic, but i've been having a problem with solving my 4x4x4 that maybe someone can help me with. I solve for the centers, match up all the middles (l's and r's) , and solve like a regular cube. I come to a situation where i have two opposing faces solved and one non-opposing face solved. That leaves only two sets of middles out of place (but still matched). I am familiar with this situation from the 3x3x3 and can solve that cube using this certain combination i learned (its one i havent seen online and have trouble describing using typical notation: the cube is held so that the two swaped middles are in the U face, then i rotate the center slice clockwise (the slice between R and L in standard notation), U, center slice, U, center slice, U", center slice counter clockwise, U, center slice counter clockwise, U, center slice counter clockwise, U"). However when i apply the same combination to my 4x4x4 it only rarely works, but it never messes the cube up either. I'm looking for someone who can explain what is happening or who can offer a substitute sequence to try.