I have known centres to prefer the side that they feel more comfortable passing and setting up their winger, over the side they carry the ball in .... most right handed people feel their passing to the left is stronger and more accurate, and therefore prefer playing on the left side ... and you can change the arm carrying the ball mid stride but you can't change the side you feel better at passing too ... having said that a good player can make more than just a fist at both sides ..
And of course there are some players that just don't have a fend ..at all ..