A Gallery of Groups · Medium problem

$D_{16}$ has order $32$, generated by a rotation $r$ of order $16$ and a reflection $s$ with $srs^{-1}=r^{-1}$. For how many rotations $r^k$ ($0\le k<16$) does $s$ actually commute with $r^k$ (that is, $sr^k=r^ks$)?