Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in Sn and An. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in Sn and An. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint 2-subsets of [n], and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in An. We prove explicit combinatorial formulas for symmetric difference-closed sets of these forms, and we prove a number of conjectured properties related to such sets which had previously been discovered experimentally using the On-Line Encyclopedia of Integer Sequences.