  In this paper we study the cohomology of smooth projective complex surfaces
$S$ of general type with invariants $p_g = q = 2$ and surjective Albanese
morphism. We show that on a Hodge-theoretic level, the cohomology is described
by the cohomology of the Albanese variety and a K3 surface $X$ that we call the
K3 partner of $S$. Furthermore, we show that in suitable cases we can
geometrically construct the K3 partner $X$ and an algebraic correspondence in
$S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the
Tate and Mumford-Tate conjectures for those surfaces $S$ that lie in connected
components of the Gieseker moduli space that contain a product-quotient
surface.
