What structure does the alternating group preserve? A common way to define a group is as the group of structure-preserving transformations on some structured set.  For example, the symmetric group on a set $X$ preserves no structure: or, in other words, it preserves only the structure of being a set.  When $X$ is finite, what structure can the alternating group be said to preserve?
As a way of making the question precise, is there a natural definition of a category $C$ equipped with a faithful functor to $\text{FinSet}$ such that the skeleton of the underlying groupoid of $C$ is the groupoid with objects $X_n$ such that $\text{Aut}(X_n) \simeq A_n$?  
Edit:  I've been looking for a purely combinatorial answer, but upon reflection a geometric answer might be more appropriate.  If someone can provide a convincing argument why a geometric answer is more natural than a combinatorial answer I will be happy to accept that answer (or Omar's answer).

Let us define an orientation of a finite set recursively.


*

*An orientation of the empty set is either $-1$ or $+1$.

*An orientation of a finite set $X$ is the data of an orientation for each $Y ⊆ X$ of cocardinality 1 (meaning $|Y|=|X|-1$) and such that for each $Z ⊆ X$ of cocardinality 2, if $X = Z∪\{a,b\}$, then the two orientations induced on $Z$ by going through the two possible paths $X → Z∪\{a\} → Z$ and $X→Z∪\{b\}→Z$ are different ("opposites").


We obtain, for each set, two possible orientations. This can be proved by showing that for any $Y ⊆ X$ of cocardinality $1$, there is a unique way of extending an orientation of $Y$ to an orientation of $X$. This also shows that each linear ordering of $X$ gives an orientation.
We can visualize what an orientation is on a drawing of a triangle or of a tetrahedron: we orient the faces/edges and make it so that everything glues nicely along the edges/vertices.
Each permutation of $X$ acts naturally on the set of orientations of $X$, and the permutations acting trivially are the even ones.
But it is not clear enough to me how this interacts in a more formal way with the intuitions there are in simplicial homology or about the exterior algebra.
