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) = NCop.
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 Kn = F*Δ[n].
Since F* preserves limits, we know that K0 = 1 (terminal object.)
For all n, there is a monomorphism Δ[n] -> Δn (n-fold product), and we can use this to regard Kn as a subobject of (K1)n.
Finally, you can get Δ[n], for n>2, as an inverse limit of a diagram involving Δ, Δ, and/or products thereof.
Thus, the functor F is basically determined once you know the simplicial set K1 and the subobject K2 of (K1)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 K1, and the linear order is the "relation" K2 on K1. This fact discussed, for instance, in Mac Lane & Moerdijk, Sheaves in Geometry and Logic.)