This is an open-ended question. If you'll allow, I'd like to keep its origins vague for the moment, so as not to bias responses. I am interested in any and all thoughts.
There are three pairings of the numbers 0,1,2,3: $\{ 01, 23 \}, \{ 02,13 \}, \{03,12 \}$. Sum the pairs and look at the parities: 0+1 and 2+3 are odd, 0+2 and 1+3 are even, 0+3 and 1+2 are odd. So let's say there are two "odd pairings" and one "even pairing."
Now draw a 3-simplex, label its vertices 0, 1, 2, 3, and observe that the above pairings are pairs of opposite edges (edges not sharing a vertex). So there is an asymmetry to the types of opposite edge pairs that a labelled 3-simplex has: it has two "odd edge pairs" and only one "even edge pair."
My question is this: Can you see some "geometric characteristic" that distinguishes the odd and even edge pairs?
Another way of asking would be: Does anything about the way that the 3-simplex sits in $\mathbb R^3$ separate its edge pairs into two kinds of pairs, similar to the above separation into "even pairs" and "odd pairs?"
I should mention that I am thinking of the vertex labeling as a way of orienting the simplex, and so I am particularly interested in responses that somehow involve orientations (but all thoughts are welcome).
Edit: Here is a better way to describe what I'm looking for. As pointed out in the comments, the bare simplex all on its own has no edges nor pairs of edges singled out; it is symmetrical in that regard. A labeling of vertices provides us one way of finding a distinguished pair of opposite edges. What are some other additional structures one can impose on the 3-simplex that would single out a pair of opposite edges? For example, choosing an orientation is a bit less structure than labeling the vertices--is an orientation sufficient to single out a pair of opposite edges?
We argue in ${\mathbb Z}/(2{\mathbb Z})=\{0,1\}$. Up to symmetry there is just one way to label two vertices of a regular tetrahedron with $0$ and two vertices with $1$. Then there are two pairs of opposite sides having sum $1$ and one pair of opposite sides having sum $0$. There is not more to it.