Let $M$ be a compact, connected, orientable surface and $\varphi_1,\varphi_2$ be two orientation-reversing involutions (i.e., diffeomorphisms for which $\varphi^2=Id$) such that the fixed-point set of both is non-empty. I am trying to understand what conditions guarantee the existence of an equivariant self-diffeomorphism $f$. (I.e., such that $f \circ \varphi_1 = \varphi_2 \circ f$, which is what I call "equivalen[ce]" in the title.)

The fixed-point set is a union of circles. Perhaps conditions can be derived from how those circles from $\varphi_1$ and $\varphi_2$ interact homologically. (More precisely, how the image of their fundamental classes through the inclusion are related to one another. Of course, necessary conditions can arise from this.) Since such a diffeomorphism restricts to a diffeomorphism between the fixed point sets, it is clear that this self-diffeomorphism does not always exist. (E.g., take a $2$-genus surface on $\mathbb{R}^3$ to be lied down, and consider reflection through the middle vertical circle as $\varphi_1$ and through the $xy$ plane as $\varphi_2$. The fixed point sets have a different number of connected components.)

I have two main questions:

- Under what conditions on $M$, $\varphi_1$ and $\varphi_2$ (or perhaps their fixed point sets) does this hold?
- If the above is a little too broad, then is it known if this always holds for $S^2$? What about $S^1 \times S^1$?