I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the *flipping lexicographic* ordering, for evident reasons. I could also imagine it getting called the *parity lexicographic* ordering, but a brief search suggests that that’s used for some slightly different orderings.
$\newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \newcommand{\N}{\mathbb{N}} \newcommand{\fl}{\mathrm{fl}} \newcommand{\lfl}{\;\sqsubset^\fl\;}$

For sets $\x, \y \in \binom{\N}{m+1}$, write $\x = \{x_0 < \ldots < x_m\}$, $\y = \{y_0 < \ldots < y_m\}$.

**Definition.** $\x \lfl \y$ if $\x$ and $\y$ differ first in the $i$th place, and

- $i$ is even, and $x_i < y_i$; or
- $i$ is odd, and $y_i < x_i$. (This is the flip!)

As for ordinary lex, there’s also a nice inductive characterisation: Write $\x = \{x_0\} \cup \x^{\geq 1}$, and $\y = \{y_0\} \cup \y^{\geq 1}$, similarly to above. Then $\x \lfl \y$ if and only if either $x_0 < y_0$, or $x_0 = y_0$ and $\y^{\geq 1} \lfl \x^{\geq 1}$. (Again, note the flip.)

Does this ring any bells with anybody?

(Of course, $\lfl$ has obvious generalisations beyond $\binom{\N}{m+1}$; I’m sticking to that case here partly for definiteness, mainly since that’s the specific case I’m interested in.)

**Background:** I’ve been playing around with implementing the algorithms from Ross Street’s “The Algebra of Oriented Simplices” (and related papers) in Haskell/Agda, and this ordering turns out to make a computationally convenient stand-in for his $\lhd$ order, in places.