The way I intended it, it does, but it doesn't make much difference since you could easily turn a polygon tiled multiple-to-a-face (if one existed) to a polygon tiled one-to-a-face by adding vertices between the quadrilaterals on the polygon faces, and making the polygon faces bend back and forth at those vertices to differentiate them.
The reason I'm posting this here is not just to discuss the problem, but to ask what I can do with this now that I have it. Is it trivial by some theory--where should I go to find that out--and if it doesn't seem to be trivial is there a next step? Find a professor? Could I possibly get it published? My proof is centered around a graphical case analysis that, while I believe is sufficient, I don't know how to make much more rigorous/algebraic.