Note: to clarify, throughout the text "faces" actually mean $(n-1)$-faces or facets in standard terminology.
In $\Bbb R^n$, let $A:=\text{simp}(s_{1,2,\cdots,n})$ be a $(n-1)$ simplex that is non-degenerate (i.e. $\dim A = n-1$) and such that $A$ isn't coplanar with $0$ (i.e. don't lie in the same $(n-1)$-affine hull with the origin). Is there an explicity way to identify the symmetric faces of $\text{conv}(A\cup-A)$?
For example, in $\Bbb R^3$, the convex hull is easily enumerated as (by drawing a picture or whatever):
1 2 3
1 3 5
1 5 6
1 6 2
2 4 3
2 6 4
3 4 5
4 6 5
And we're readily aware of the symmetric pairs.
However, in higher dimensions, is there a way to easily enumerate all faces of the convex hull and identify symmetric pairs?