Yes, most bubs do sort of get themselves into position - usually head down, facing your back, head slightly turned to the left or right. (On the chart, it would be marked LOA - left occiput anterior, or ROA - right occiput anterior.) Those are the most common presentations, and the easiest to deliver, as the PP mentioned. That presents the smallest head circumference. Some babies are born looking up, or to the side, which is a more difficult delivery, and some are "face" presentations, which is also quite difficult, because the face doesn't mould the way the skull of a newborn will. (Is that the right "mould"? hmmmm, it's looking funny to me. Oh well, you know what I mean. ) The mother would need to have quite a roomy pelvis for that to work, as the pp said.
Anyways, yes, they do usually get into position, but anything can happen, and sometimes they will need a little guidance. It probably didn't have anything to do with you pushing too early.