Characterizing covers via simple closed curves

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 Teichmuller 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.