Let $k$ be an integer. Traditionally a partition $\pi=V_1\cup \dots \cup V_n$ of the set $[k]:=\{1,\dots, k\}$ is called crossing when there exist $a,c\in V_i$ and $b,d\in V_j\not= V_i$ such that $a<b<c<d$. For example, the partition $\{1,3\}\cup\{2,4,5\}$ (which, for convenience, I will just denote by $(12122)$) is a crossing partition of $[5]$ whereas $(21122)$ is a non-crossing partition.

For my needs I am interested in a different notion of crossing, namely only those crossings that can't be resolved in an even number of permutations. For example, in my stricter notion of crossing $(1223123)$ would not have a crossing between $1$ and $2$ since for resolving this crossing, we would have to move $1$ past an even number of $2$'s. It would, however, still have crossings between $1$,$3$ and $2$,$3$.

Is this notion *known*? Do there exist results on the combinatorics of this stricter notion of crossing?