  We describe new relations among conformal block divisors in
$\operatorname{Pic}(\bar{\operatorname{M}}_{0,n})$. These relations appear from
various rank-level dualities of conformal blocks on $\mathbb{P}^1$ with $n$
marked points. We also give a coordinate free description of rank-level duality
maps on $\overline{\mathcal{M}}_{g,n}$.
