That is the question in essence. The first definition of orientability is the following:
A regular surface $S$ is called orientable if it is possible to cover it with a family of coordinate neighborhoods in such a way that if a point $p\in S$ belongs to two neigborhoods of this family, then the change of coordinates has positive Jacobian at $p$. The choice of such a family is called an orientation of $S$, and $S$, in this case, is called oriented. If such a choice is not possible, the surface is called nonorientable.
Well, what if we cover $S$ with only one coordinate neigborhood? Then (I think) the surface is always orientable and this would be a global property of the surface. This is what my intuition tells me, but how do I formalize this argument?