Let G=(V(G),E(G)) be a graph and let A=(A,+) be an abelian group with identity 0. Then an A-magic labeling of G is a function ϕ from E(G) into A∖{0} such that for some a∈A, ∑e∈E(v)ϕ(e)=a for every v∈V(G), where E(v) is the set of edges incident to v. If ϕ exists such that a=0, then G is zero-sum A-magic. Let G be the cartesian product of two or more graphs. We establish that G is zero-sum ℤ-magic and we introduce a graph invariant j∗(G) to explore the zero-sum integer-magic spectrum (or null space) of G. For certain G, we establish A(G), the set of nontrivial abelian groups for which G is zero-sum group-magic. Particular attention is given to A(G) for regular G, odd/even G, and G isomorphic to a product of paths.