Is there a name for the set of the pair/impair equivalence classes of the permutations of a set?

Question

In the theory of abstract simplicial complexes, a face of an abstract simplicial comples is a set, while an oriented face is one of the two equivalence classes of the permutations of this set under the equivelence relation based on the parity of the number of inversions. Is there a name for these equivalence classes?

Edit:

Sorry, I posed my question badly. Really, I am looking for a name of the set of these two equivalence classes. E.g., if the desired name is somename, then I'd like to say, that "Given a set $S$, an elements of somename of $S$ is called an oriented face."

Edit 2

Or, still it is good also, if the name refers to the equivalence classes itself, but simmetrically. E..g. "The othersomenames of $S$ are called oriented faces".

(The title also changed)

Answer 1

So one of the classes is called "alternating," and I guess the other could be called "non-alternating." Or, alternately, "odd" and "even."

Edit: So you're essentially asking of there's a specific name for the group $S_n / A_n$? I haven't heard of one; usually people call it by its isomorphism class, $C_2$.

It would follow a general pattern to refer to "a permutation, up to an even permutation," but that sounds horrible. "A parity of permutations" would perhaps be a bit better.