It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a primitive embedding $\iota : M \to Pic(S)$ that contains a pseudo-ample class, we call $S$ an $M$-polarized K3 surface.
Mirror symmetry for this case essentially comes from splitting $H^2(S)$ into $M$ and (in a certain sense) an orthogonal complement, which yields a mirror K3 which is $M^\vee$-polarized for a certain lattice $M^\vee$ of rank $20 - r$.
Q1: Is there a similar story for abelian surfaces?
We have another construction that we can produce from abelian surfaces. Given an abelian surface $A$ with Picard rank $r$, there is a canonically associated K3 surface (see, e.g., "Modular Invariants for Lattice Polarized K3 Surfaces", A. Clinger and C. Doran), the Shioda-Inose K3 $SI(A)$ whose Picard rank is $ 16 + r$ (this K3 is birational to a double cover of $Km(A)$, the Kummer surface of $A$). The nice fact about this K3 is that it is Hodge-theoretically isomorphic to $A$.
So let's start with our abelian surface $A$. Assuming the answer to the first question is positive, then we have a mirror abelian surface $A^\vee$ whose Picard rank is $4 - r$. We also have the chain of relationships $$ A \leadsto SI(A) \leadsto SI(A)^\vee $$ which also has Picard rank $4 - r$.
Q2: Again, assuming a positive answer to the first question, is there some relationship between $A^\vee$ and $SI(A)^\vee$? The latter is obviously not the Shioda-Inose K3 of $A^\vee$ since its Picard rank is too small. But is there some other relationship there?
