One player starts with the round dial and every turn that player turns the dial to the next digit and hands it over to the other player.
The only situation ive found in the rules when this has any significance is in the command phase when both players play a card of equal priority. When this happens, the player with the round marker rolls a red defense die to determine who gets priority for that round. A dieroll that can be done by either player and aside from some form of etiquette I cant see it matter at all.
Tracking rounds through use of the dial is all good and whatnot, many will probably have their own form of dials or markers to track rounds anyway, but handing it back and forth seems completely unnecessary to me.
Am I missing something that makes it important or can it just as easily be left out?