Probably the right thing to do is to express the classifying space of $A_n$ as the non-trivial double cover of the classifying space of $S_n$. A point in the classifying space is then a set of $n$ points in $\mathbb{R}^n$ \mathbb{R}^\infty$with a "sign ordering". A sign ordering is an equivalence class of orderings of the points, i.e., ways to number them from 1 to$n$, up to even permutations. I coined the term "sign ordering" by analogy with a cyclic ordering. But that name aside, the idea comes up all the time in various guises. For instance an orientation of a simplex is by definition a sign ordering of its vertices. This is in the same vein as your other examples and you can of course do something similar with any subgroup$G \subseteq S_n$. You can always choose an ordering of the points up to relabeling by an element of$G$. A bit more whimsically, you could call the configuration space of$n$sign-ordered points in a manifold "the configuration space of$n$fermions". Although a stricter model of the$n$fermions is the local system or flat line bundle on$n$unordered points, in which the holonomy negates the fiber when it induces an odd permutation of the points. This local system is similar to the sign-ordered space in the sense that the sign-ordered space is the associated principal bundle with structure group$C_2$. 1 Probably the right thing to do is to express the classifying space of$A_n$as the non-trivial double cover of the classifying space of$S_n$. A point in the classifying space is then a set of$n$points in$\mathbb{R}^n$with a "sign ordering". A sign ordering is an equivalence class of orderings of the points, i.e., ways to number them from 1 to$n$, up to even permutations. I coined the term "sign ordering" by analogy with a cyclic ordering. But that name aside, the idea comes up all the time in various guises. For instance an orientation of a simplex is by definition a sign ordering of its vertices. This is in the same vein as your other examples and you can of course do something similar with any subgroup$G \subseteq S_n$. You can always choose an ordering of the points up to relabeling by an element of$G\$.