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 $(3,L,d\mathrm{/}6)$. We also explain for which $d$ the Hilbert schemes ${\text{Hilb}}^n(S)$ and ${\text{Hilb}}^n(S^{\tau })$ are birational.