I don't know such a classification, though I'm interested. Another standard endofunctor is op:Δ -> Δ which is the identity on objects but which relabels morphisms by reversing the sense of the ordering in each set [n].

Thus, if NC is the nerve of a category, op(NC) = NC^{op}.

Edit: Here is a thought which might lead to a classification.

Given an endofunctor F: Δ->Δ, there is a restriction functor F^{*}: S->S,
where S=Psh(Δ, Set) = simplical sets. This has a left adjoint F_{#}, which on representable presheaves is isomorphic to the original functor F. So F is determined by F_{#}, which is determined by the value of F^{*} on representables.

Write K_{n} = F^{*}Δ[n].

Since F^{*} preserves limits, we know that K_{0} = 1 (terminal object.)
For all n, there is a monomorphism Δ[n] -> Δ[1]^{n} (n-fold product), and we can use this to regard K_{n} as a subobject of (K_{1})^{n}.

Finally, you can get Δ[n], for n>2, as an inverse limit of a diagram involving Δ[1], Δ[2], and/or products thereof.

Thus, the functor F is basically determined once you know the simplicial set K_{1} and the subobject K_{2} of (K_{1})^{2}.

(More is true. In the above, what I'm really doing is using the fact that S is the *classifying topos* for linear orders. In other words, adjoint pairs G: S <==> E: H
where E is a topos, and the left adjoint G preserves finite limits, correspond to "objects in E equipped with a linear order". In this case, E=S, and G=F^{*}, the object of E with a linear order is K_{1}, and the linear order is the "relation" K_{2} on K_{1}. This fact discussed, for instance, in Mac Lane & Moerdijk, *Sheaves in Geometry and Logic*.)