Is there a theory of endofunctors of the category $\Delta$ of finite linear orders ? Can they be classified ? Is there a reference on this ?

Can one classify endofunctors $T:\Delta\to\Delta$ which give rise to sequences in $sC:=\operatorname{Func}(\Delta^\text{op},C)$ $$X_\bullet\to X_\bullet\circ T \to X_\bullet\times X_\bullet$$ (maybe with some nice properties making $X\circ T$ a path object etc.)?

The motivation for the question is that these endofunctors are sometimes used to define simplicial analogues of path spaces. I am particularly interested in the "edgewise subdivision" endofunctor, as explained in Dmitri Pavlov's answer. I quote two comments about this endofunctor.

This endofunctor is basically a special case of the simplicial join construction. See kerodon.net/tag/016K and also ncatlab.org/nlab/show/join+of+simplicial+sets. — Dmitri Pavlov

The endofunctor of simplicial sets induced by precomposition with [x2] on \Delta is the case r=2 of the r-fold "edgewise subdivision". See section 1 of Bokstedt, Hsiang, Madsen, Invent. Math. 111 (465-540) 1993, or section 6.2.1 of Dundas-Goodwillie-McCarthy, "The local structure of algebraic K-theory". A conceptual precursor is given in appendix 1 of Segal, Invent. Math. 21 (213-221) 1973, with the idea being credited to Quillen. — John Rognes