If I interpret your question correctly, you are asking which class in $H^1(S; \mathbb{Z}/2 \mathbb{Z})$ corresponds to the orientation covering. This is the first Stiefel-Whitney class of $TS$, and there are [many constructions][1] for it.


  [1]: https://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class