  Given two finite covers $p: X \to S$ and $q: Y \to S$ of a connected,
oriented, closed surface $S$ of genus at least $2$, we attempt to characterize
the equivalence of $p$ and $q$ in terms of which curves lift to simple curves.
Using Teichm\"uller theory and the complex of curves, we show that two regular
covers $p$ and $q$ are equivalent if for any closed curve $\gamma \subset S$,
$\gamma$ lifts to a simple closed curve on $X$ if and only if it does to $Y$.
When the covers are abelian, we also give a characterization of equivalence in
terms of which powers of simple closed curves lift to closed curves.
