Using geometry and a cell structure derived from spaces of unlabeled configurations of points in Euclidean space, we compute the mod-2 cohomology of the collection of all symmetric groups as a Hopf ring. The Hopf ring structure we find includes a diagrammatic basis whose elements have explicit geometric interpretations, and both products (along with the action of the Steenrod algebra) are explicitly described in terms of these basis elements. Time permitting, we will discuss work in progress extending these results to alternating groups and computations at other primes. This is joint work with Paolo Salvatore and Dev Sinha.