Indeed, the Moon does look ‘upside down’ in the Southern Hemisphere compared to the northern hemisphere. This is simply a matter of orientation.
Imagine if the Moon orbited in the same plane as the equator. If you were in the northern hemisphere, the Moon would always appear in the southern sky since that is the direction of the equator. The reverse is true in the southern hemisphere: the Moon would appear in the northern sky.
So, these two observers are looking at the same object from opposite directions and naturally that means one sees the object flipped compared to the other. This means that the ‘Man in the Moon’ is upside down in the southern hemisphere, and can actually look more like a rabbit.