In the southern hemisphere, does the Moon appear to start its waxing phase in reverse?
It depends on what you mean by "reversed". The east-west directions remain the same but the left-right directions are reversed.
From one day to the next, the Moon shifts eastward relative to the Sun no matter where you are on Earth. In its waxing phases the Moon has moved east of the Sun and its lighted area, pointing towards the Sun, is on the west side. When the Moon is waning and coming towards the Sun from the west the lighted area is on the east side.
The Moon orbits near the extension of the Equator into space, so viewers in the Northern Hemisphere are generally looking southward while those in the Southern Hemisphere are looking northward. Since the east-west directions remain the same as we saw above, the left-right directions must be reversed.