A Gallery of Groups · Hard problem

Represent $D_{15}$ (order $30$) by maps $x\mapsto ax+b\pmod{15}$ with $a\in\{1,-1\}$, so that $r\colon x\mapsto x+1$ and the reflection $s\colon x\mapsto-x$ (as in the lesson's proof). How many elements of $D_{15}$ commute with $s$?