Footman described the problem of disappearing polygons in general terms
I haven't encountered it myself, and certainly not in such a critical form as here.
But I think the first thing to do is to check the UV mapping. See if it breaks, and if so, at what stage.
P.S.
Waterfall might also be right, and the problem might be with flipped normals. To check this, you can simply apply a two-sided material to the model.