Track stands! Not a contradiction to your statement at all though: you need to be moving just ever so slightly.

With a fixie it's easy, because you can pedal forwards and backwards in tiny amounts. With a freewheel, it's trickier but you get the hang of it with practice. Ideally you'll have an incline, so you pedal forward to go forward, and ease up to slide back. After some practice I can use the raised reflective paint from e.g. crosswalks as the "incline." This miniscule motion is enough to balance

and like you said, it ain't the angular momentum that does it.