Hodge structure on K3 surface (the middle line of Hodge diamond is 1 20 1) is similar to the Hodge structure of cubic fourfold (the middle line of Hodge diamond of primitive cohomology is 0 1 20 1 0). But the two Hodge structures have different signature. To make the signatures the same, we may remove a class of polarization from $H^2(S)$ and an extra class from $H^4_{prim}(X)$.
We will consider a cubic fourfold $X$ with a hyperplane class $h\in H^{1,1}(X)$ and a nontrivial class $T$ in $H^{2}_{prim}(X,\mathbb Z)$, in this case we put $d=\mathrm{disc}(h^2, T)$ is the discriminant of lattice generated by $T$ and $h^2$ (it is a sublattice of $H^{4}(X, \mathbb Z)$).
Question. Are there examples (besides two given below) of geometric constructions (for example, a construction of a cycle in $X\times S$) of isometries between $H^4_{prim}(X)^{\bot T}$ and $H^2_{prim}(S)$ for some cubic fourfold $X$ and a K3 surface $S$?
Example $d=6$. Fourfold $X$ contains a double point $p$ and projection from point $p$ defines a birational map from $X$ to $\mathbb P^3$ and a regular map $$ Bl_p(X)\to\mathbb P^4, $$ which contracts a K3 surface $S$.
Example $d=14$ (Beauville-Donagi). Let $P=\mathbb P(\Lambda^2 \mathbb C^6)$ be 14-dimesional projective space of skew two-forms. Plucker embedding gives $$ Gr(2,6)\subset P. $$ In the dual space $P^*$ of dual forms we have a variety of degenerated (i.e., of rang 4 or less) skew 2-forms: $$ Pf(4,6)\subset P^*. $$ Obtaining $X$ as an intersection of $Pf(4,6)$ with a general $\mathbb P^5\subset P^*$, K3 surface will be given by the intersection of $Gr(2,6)$ with $(\mathbb P^5)^\bot\subset P$. It is not difficult to construct a correspondence (and Fourier-Mukai kernel) for $X$ and $S$.
There is a non-example for $d=8$, it has deal with $X$ and a K3 surface with a distinguished Brauer class. Also the following theorem holds.
Theorem (Addington-Thomas). If $X$ is as above with $d$ satisfying some numerical condition ($d=6,14,26,38\ldots$), then there exist a polarized $K3$ surface $S$ of degree $d$ and an algebraic cycle in $A^3(S\times X)_{\mathbb Q}$ which induces a Hodge isometry $H^2_{prim}(S,\mathbb Z)\to(h^2,T)^{\bot}$.