I see your problem: You are verbally equating changes in the viewers' perspectives as differences in "the side of the Moon".
A more realistic statement is that: People viewing the Moon from the Earth's South Pole see the same view of the Moon as people viewing it from Earth's North Pole, but each would interpret the other's view as "upside down", because the viewers themselves have defined "down" as 180 degrees different.
So, photographs taken by those two viewers and rotated by how ever many degrees are needed to make both "downs" coincide with the local "down" anywhere would appear to show opposite orientations of the same side of the Moon, thus appearing to make a complete circle in the combined photo that matches the local horizantals instead of the moom images. Of course, that would be obvious if the pictures had any detail of the actual Moon surface.
A similar argument can be made for two observers on different sides of the Earth at the Equator, but only if one is taking the picture at local moonrise and the other at local moon set. If they both do it at their local rise or set times, the pictures will have the same orientation of the illuminated cusps of the Moon image with respect to their local horizons.
But, I guess that you are thinking the confusing wording is OK content for an article about a mumbo-jumbo subject. The problem with that here is that this particular venue is supposed to be as firmly based on actual science as we can make it. Yes, the people who believe in the mumbo jumbo subject would speak like the article's text, but the purpose of the article seemed to be to put that into a more science-astronomy perspective.