earth - Where exactly does the Moon flip, given that it appears the other way up in the other hemisphere?

This doesn't really answer your question, but the orientation of the moon is changing constantly, because we tend to compare the moon to the horizon, and think of the horizon as fixed. Here's a video demonstrating the effect (simulated) for the full moon over London on August 28-29 (all times GMT):

https://www.youtube.com/watch?v=QXVpRfaJXmY

If you were changing location, you would see a similar gradual effect.

The only real place you would see a true "flip" is near where the moon is at zenith (overhead), because, as you passed the zenith point, the moon may go from being 89.999 degrees high in the western sky to 89.999 degrees high in the eastern sky (and flip directions because you're now comparing it to a different horizon).

Realistically, when the moon is that high, you probably wouldn't be comparing it to the nearest horizon.