Heterotic/Type IIA Duality on K3 and $T^{4}$

Question

One of the many interesting dualities in string theory emerging in the 90s is a duality between Type IIA string theory compactified on a K3 surface and heterotic string theory compactified on a four-torus $T^{4}$.  One can see this in the brief survey of Aspinwall (https://arxiv.org/pdf/hep-th/9508154.pdf).  

Perhaps my question is a more general one about dualities, but I'm hoping someone can outline exactly what mathematical structure on one side corresponds to something nice on the other side.  More concretely, imagine I'm looking at a very special four-torus $E \times E$ of an elliptic curve times itself.  This one has three algebraic curve classes, call them $E_{1}, E_{2}$ corresponding to the two factors of $E$, and a diagonal class $\Delta$.  One can show that the intersection form of this special abelian surface is the quadratic form 

$$Q(x,y,z) = xy + yz+xz.$$

Now what, if anything, will this quadratic form translate into on the K3 side under the duality described above?  Will it determine some special K3 with a particular intersection form which is dual to my $E \times E$?  To give some motivation for my question, I'm studying certain topics in topological string theory on abelian surface fibrations.  Certain equations are extremely reminiscient of how the story unfolds in the K3 case, and I'm wondering if some physical string theory duality is underlying it.