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 $d$, each associated K3 surface $(S,L)$ canonically yields another one, $(S^{\tau},L^{\tau})$. We prove that $S^{\tau}$ is isomorphic to the moduli space of stable coherent sheaves on $S$ with Mukai vector $(3,L,d/6)$. We also explain for which $d$ the Hilbert schemes $\text{Hilb}^n(S)$ and $\text{Hilb}^n(S^{\tau})$ are birational.