After many hours of troubleshooting I finally found a solution to having a 2 sided cloth in Cinema 4D with thickness. And it was easy.
With thickness set to 0 in the cloth nurb the textures map correctly, but the second I added a thickness greater than 0 it mapped 1 texture for both sides. To get around this I tried a thin rectangle to simulate thickness. It kind of worked, but even with self collision on the back of the cube goes through the front resulting in a weird morphing texture jumble. I also found another workaround using a proximal shader, but this is less than ideal and the level of control with this method is not even worth it.
Plane in Cloth Nurb with thickness at 0
Plane in Cloth Nurb with thickness at 1
The Solution? Believe it or not, negative thickness. If you set the thickness of the cloth to be -1 instead of 1 it just works for some reason. A great example of a strange problem with an even stranger solution.
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