When do Sylow $p$ and Sylow $q$ subgroups commute?

Question

Do $p$-Sylow and $q$-Sylow subgroups commute iff both are unique and thus normal? I know that one direction is true: namely that if the $p$-Sylow subgroup and the $q$-Sylow subgroup are normal in the group, that they commute.

My question: Is the other direction also true?

Answer 1

Counterexample: $G=S_3$, $p=2$ and $q=3$.