A friend asked me for the time. When I looked at my analog clock (which only has a minute hand and an hour hand, no second hand), I thought of telling her the time with the hour and minute hands swapped, and then have her work out the actual time. Then I realized that not all clock-hand configurations make sense when the hands are swapped.
Let $T$ denote the set of all clock-hand configurations on an analog clock that occur over twelve hours.
Question: What is the subset of times $S\subseteq T$ such that when the hands are swapped, the time still makes sense? (In other words, $S$ is the maximal subset of $T$ closed under the operation of swapping the hands.)
Comments: $S\neq T$, since for example $3:00\notin S$. But $S$ is non-empty, since $12:00\in S$.