"Skew Cohomology" of a Space Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$.  The action is induced
by permuting the vertices. 
Let $\Sigma_{n+1}$ act on $\Bbb Z$ by means of the sign representation.
Then the singular $n$-cochains 
$$
S^n(X) := \text{map}(X^{\Delta^n},\Bbb Z)
$$
inherits a $\Sigma_{n+1}$-action given by conjugating functions.   
Definition. 
The skew $n$-cochains on $X$ is given by the invariants 
$$
S^n(X)^{\Sigma_{n+1}}
$$
that is, by equivariant functions $X^{\Delta^n} \to \Bbb Z$.
Then an elementary calculation shows that the usual singular coboundary operator $\delta$
maps $S^n(X)^{\Sigma_{n+1}}$ into $S^{n+1}(X)^{\Sigma_{n+2}}$.
So we get a cochain complex:
$$
S^0(X) \overset \delta \to \cdots \overset \delta \to S^n(X)^{\Sigma_{n+1}} \overset \delta \to S^{n+1}(X)^{\Sigma_{n+2}} \overset \delta \to \cdots
$$
Define the skew cohomology of $X$ to be the cohomology of this cochain complex.
Questions: What is it? What properties does it have? Has it ever before been studied?
 A: I do not have the book at hand to check, but maybe you do...
Loday defines in his book on Cyclic homology what a crossed simplicial group is and for each such gadget a corresponding homology theory. There is, for example, the "cyclic" crossed simplicial group, and the corresponding homology theory is cyclic homology. Well, there is a crossed simplicial groups---call it $\Delta S$---built from symmetric groups, and it turns out that the corresponding homology theory coincides with Hochschild homology. I am pretty sure your complex is the analogue of Connes' $\lambda$-complex corresponding to the homology theory for $\Delta S$, which is quasi-isomorphic to the complex which defines the actual $\Delta S$-homology rationally and, therefore, should just give you simplicial homology over $\mathbb Q$.
Integrally, you should not take invariants in each degree, but consider a double complex, each of whose columns are the complex which computes $H^\bullet(S_{n+1},\mathord-)$, much as the double complex which computes cyclic homology has columns which compute $H^\bullet(C_{n+1},\mathord-)$.
(I would love to have an example where taking invariants integrally does not work, by the way!)
A: I believe it should be exactly the same as ordinary singular cohomology.  It should define a cohomology theory for the exact same reason that usual singular cohomology does (the usual proof of excision by subdivision seems to work since the cosubdivision of a $\Sigma_n$-invariant cochain can be checked to still be $\Sigma_n$-invariant; the usual proof of homotopy invariance doesn't work because the usual triangulation of a product $\Delta^n\times I$ is incompatible with the $\Sigma_n$ action, but all you need to do is geometrically construct a different, finer triangulation that is more symmetric), and on a point it can be computed explicitly.
As a heuristic argument for why this should be true at least rationally, note that de Rham cohomology already has this $\Sigma_n$-invariance built in, in the requirement that differential forms be alternating.
