Wouldn’t that effect be the same on a flat earth or a spherical earth? The top of the visible moon rises first but sets last? Is that the OP is calling an inversion?
Spherical Earth/Heliocentric model:
This more or less is an illusion. Not the moon tilts or turns, it's the observer, who changes orientation.
When viewed from the reference system "equatorial plane" or North/South Pole, the moon does not "turn". Astronomers use an Equatorial Mount to take images of the moon.
The observers "natural" reference system is defined by gravity and the horizon. If you are looking west or east to the horizon, this horizon is not parallel to the equatorial plane. It's tilted by an angle given by latitude: 90°-lat. In the south the horizon is parallel to the equatorial plane. So on his journey from rise/east to south the moon tilts by 90°-lat. and the same from south to set/west.
So, when the moon rises in the east and sets in the west, the tilt is 2*(90°-lat).
Only on the equator this could be 180° - "top" of moon rises first and sets last.