Where exactly does the Moon flip, given that it appears the other way up in the other hemisphere?
The moon doesn't flip. Is us who flip.

Those in the antipodes have their heads pointing in the opposite direction. So the moon appears inverted. But there is no sudden point where it "flips". We don't live on a discworld.

You can see this yourself. Because as the Earth spins you will change the direction in which your head is pointing. Look at the full moon over the course of one night. As it appears to travel from East to West in the sky, it's orientation will slowly change. There is no mystery here. You are being rotated by the Earth.