Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms modulo isotopy. Homeomorphisms are allowed to **reverse orientation** or **permute** boundary components.

- What can one say about its torsion elements?
- Is it true that every mapping class of order 2 is orientation reversing?

Now assume that $S$ also has marked points on the boundary, and homeomorphisms are also allowed to permute them.

- Is it true that every mapping class of order 2 is orientation reversing?

It is well-known that if a mapping class preserves orientation and fixes each point of $\partial S$, then it cannot have finite order. It is not clear to me what happens when we allow orientation reversing or permutations.

anyclosed surface andanyfinite group acting on it, and remove disks about any orbit of this group. So, examples are a plenty. $\endgroup$3more comments