We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $Δ$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arranged in a $3\times 3$ array. Then for any continuous map $f:Δ\to \mathbb{R}^3$ it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces $σ$ and $τ$ induced by the two parts satisfy $f(σ)\cap f(τ) \neq \emptyset$. Our result also has consequences for geometric transversals and topological Helly.