It could, and then the hanger would be facing the other direction, and everything would simply be mirrored left-right. The last arm to hold the hanger still has to replace it on the rack, which means it needs the hanger's hook facing away from it, which only happens when there have been an even number of exchanges between arms. Thus the need for that extra exchange.

That extra exchange could have been done after putting the shirt on the hanger, but that would have been more risk for it falling off again.