This problem emerged while trying to generalize the notion of a path homotopy.
Consider the standard $n$-simplex $\Delta^n \subset \mathbb{R}^{n+1}$. I want to give the standard labeling to it.
Here, I coin the term semifacet, which is a facet that is chosen a face it belongs to. Likewise, semiridge (which belongs to a semifacet), semipeak (which belongs to a semiridge), and semi-$k$-face are defined. As a consequence, an $n$-simplex has $n+1$ faces, $(n+1)×n$ semifacets, $(n+1)×n×(n-1)$ semiridges, and so on till $(n+1)!$ semi-$0$-faces.
Two semi-$k$-faces are said to match-up iff, they are the same $k$-face and there uniquely exists $m \in \{k,\cdots,n\}$ such that there doesn't exist an $m$-face where both semi-$k$-faces belong. Note that matching-up is symmetric, but neither reflexive nor transitive.
I label all the faces, all the semifacets, all the semiridges and so on. The standard labeling shall satisfy all the following conditions:
For all $i \in \{1,\cdots,n+1\}$, the face not intersecting the $i$-th axis, $\partial_i(\Delta^n)$, is labeled $i$.
$\partial_{n+1}(\Delta^n) = \Delta^{n-1}$ has the standard labeling.
For every two matching-up semi-$k$-faces, their labels sum to $k+3$.
I proved that such labeling exists on the semifacets. For every semifacet that belongs to the face labeled $i$, take $j$ as the label on the opposite face, and the desired label is $j-i \in \mathbb{Z}/(n+1)\mathbb{Z}$.
Here's how a tetrahedron will be labeled, shown as its net:
I don't know how to prove uniqueness though. And I don't know how to go through semiridges, semipeaks and so on. How?