I have two reflections $r$, $s$ with $r^{2}=s^{2}=e$.
Also $rs \neq sr$ and $(rs)^4 = (sr)^4= e$.
I know $r$ and $s$ generate a group of order $16$ but that was by writing each matrix multiplication by hand. I wanted to know if there was a simpler way of finding the order.
I thought maybe it has something to do with dihedral group $D_{16}$ but the condition $rsr^{-1} = s^{-1}$ is not satisfied with the reflections I have.
This is a presentation for the dihedral group of order 8. $rs$ is your generating rotation, and you can check that it is inverted by conjugation by $r$ or $s$. There are a few other details to check...
Note that two distinct involutions always generate a dihedral group of order twice the order of their product. (Possibly infinite.)