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. 

  [1]: https://m.facebook.com/story.php?story_fbid=10157493966588914&id=811018913