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 $S_n$ and $A_n$. 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 $S_n$ and $A_n$. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of a set consisting of pairwise disjoint 2-subsets of $[n]$. We prove an explicit combinatorial formula for the number of symmetric difference-closed 4-sets satisfying this property, and we prove an explicit combinatorial formula for an analogous class of symmetric difference-closed sets consisting of elements which correspond to commuting involutions in $A_n$.