O'Grady tenfolds as moduli spaces of sheaves

We give a lattice-theoretic characterization for a manifold of $\mathrm{O}\mathrm{G}10$ type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of $\mathrm{O}\mathrm{G}10$ type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.