Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone reordering and degeneracy
$s_i :S_k \rightarrow S_{k+1} $ by adding 1 to all elements with $\eta(j)>\eta(i)$ and incerting into the result a new element $\eta(i)+1$ right after $\eta(i)$ on $i+1$ place. It is a simplicial set, contractible and classifies reorderings of simplicial sets.

Is it known? May be in higher symmetric something?

(Update) Boris Tsygan pointed the right direction in [Facebook duscussion] 1

The object is classical and it has a name "Symmetric crossed simplicial group”. It was introduced almost simultaneously in

Appendix A10, page 191 “Symmetric objects” B. L. Feigin and B. L. Tsygan “Additive K-theory” 1987 K-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86 LNM 1289

Krasauskas, R. "Skew-simplicial groups", Lithuanian Mathematical Journal, Jan 1987 vol 27 issue 1 p. 47--54

And independently

Zbigniew Fiedorowicz and Jean-Louis Loday “Crossed simplicial groups and their associated homology” Trans. Amer. Math. Soc. 326 (1991), 57-87

It has big value in everything symmetric. Geometric realization $|S_\bullet|$ is the topological group structure on infinite dimensional sphere.