I posted this as a thread, however I was told to post it as a one answer...
This is about the 4x4(I forgot to specify in the post):
Hi, I was wondering if there is a pure dedge flip that I can use that does not mess with centers at all OR swaps them in a perpendicular to the dedge being flipped, so that you can do something like r parity r' without it messing up the centers. Sorry if there is a post somewhere about this, I couldnt find one.