Say we have three (non-coplanar) vectors $\vec{a}, \vec{b}, \vec{c} \in \mathbb{R}^3$. Geometrically they can be used to define $3$ distinct edges of a unique non-oriented 3D parallelipiped. We define two distinct oriented 3D parallelipipeds corresponding to this non-oriented one. One oriented parallelelipiped corresponds to even permutations of $\vec{a}, \vec{b}, \vec{c}$ and the other to odd permutations.
Question: Geometrically, why is it meaningful/useful to define a notion of oriented parallelipiped that is identified with $3!/2$ distinct permutations of $\vec{a}, \vec{b}, \vec{c}$? Why not one distinct "oriented" object for each of the $3!$ distinct permutations of $\vec{a}, \vec{b}, \vec{c}$?
This corresponds to how the determinant of the even permutations of $\vec{a}, \vec{b}, \vec{c}$ (when treated as columns of a $3 \times 3$ matrix) has the opposite sign of the odd permutations of $\vec{a}, \vec{b}, \vec{c}$. So an answer to the question would imply one explantion for why the determinant should be considered "most useful", rather than say some function of the three vectors which gave distinct values differing by factors of e.g. the $(3!)$th = $6$th roots of unity for each of the $3!$ distinct permutations of $\vec{a}, \vec{b}, \vec{c}$.
Please let me know if there is anything I should clarify, or whether you believe this question to be a duplicate or there to be some similar issue.
Note: This question actually generalizes to oriented $n$-dimensional volumes for arbitrary $n$ (replace $3!$ with $n!$), and conceptually my question is about the geometric interpretation of skew-symmetric algebras generally. Discussing only the determinant in $\mathbb{R}^3$ is to make the question as accessible to as many people as possible.