Everybody seems to just use these maps but there is never a formal definition given. I looked at this question: Orientation preserving homeomorphisms but no answer is given and I can't make sense of the comments.
I've never seen a formal definition of orientability either, our professor just drew a circle-arrow on the blackboard. So I'm going with the definition I found on Wikipedia at the moment: A surface has an orientation if there is a triangulation such that I can draw a circle-arrow on each triangle such that on neighboring triangles the arrows on the circle 'point in the same direction', i.e. I could continuously move one into the other. Now going from this rather intuitive intuition, what does it mean for a self-homeomorphism to be orientation-preserving? I would suspect it means something like, if I describe my circle-arrow as embeddings $\gamma:S^1\rightarrow X$ which 'grow in the same direction as the arrow on the circle points' and where X is my space, then if the image $\phi\circ\gamma$ lies in some triangle, it again 'grows in the same direction' as the arrow on the circle in that triangle points. Is this correct?