  We prove that projective hyperk\"{a}hler manifolds of K3$^{[n]}$-type
admitting a non-trivial symplectic birational self-map of finite order are
isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3
surfaces. Motivated by this result, we analyze the reflections on the movable
cone of moduli spaces of sheaves and determine when they come from a birational
involution.
