Two polarized K3 surfaces associated to the same cubic fourfold

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarized K3 surfaces of degree d via their Hodge structures. For half of the d, a generic special cubic has not one but two different associated K3 surfaces. This induces an involution on the moduli space of polarized K3 surfaces of degree d. We give a geometric description of this involution. As an application, we obtain examples of Hilbert schemes of two points on K3 surfaces that are derived equivalent but not birational.