The moon rose in the east at about half past midnight and it would obviously have been it's bottom edge that was lit (the side pointing at the sun). After arcing across the sky, at 6am it was at about it's highest point. It would have been closest to the southern horizon so that would be your reference point, i.e. you'd consider down to be towards the south, but since the lit side would be pointing further to the east now (where the sun would be rising from) you may get the impression that the moon had rotated, but really it's your frame of reference that has changed.
If you saw it well after midnight, then again at around 6am, i'd say you need to get more sleep! ;)