I don't remember where I read it and if I remember it correctly but does the following hold true?
If $M,N$ are two (smooth?) surfaces and $f: M \to N$ is a homeomorphism such that $det(J_f)$ (the determinant of the Jacobian of $f$) has constant sign then $f$ is a diffeomorphism
Since the Jacobian of $f$ has constant sign that means it is nonzero at each point. Therefore, the inverse function theorem applies and you obtain a local inverse at each point, and that restricted function is a local diffeomorphism. If this was all we had then it would not be enough because local injectivity need not imply global injectivity. However, you also have that $f$ is a homeomorphism thus $f$ is a continuous bijection with continuous inverse. I believe this means we can weave together the local diffeomorphisms to form a global diffeomorphism.