Let $\mathcal P$ be a convex polytope of dimension $d$. For each face $F\ne\varnothing$ of $\mathcal P$, we assign some arbitrary interior point $x(F)$. Let's suppose WLOG that $x(\mathcal P)=0$.
We define a flag as a chain $f=(f_0,f_1,\ldots, f_d)$ where $f_k$ is a $k$-face, and these are all pairwise incident. We say that two flags are adjacent if they share all but one face. To each flag, we assign an oriented basis $B(f)=(x(f_0), x(f_1), \ldots, x(f_{d-1}))$.
Conjecture: If $f$ and $g$ are adjacent, then $B(f)$ and $B(g)$ have opposite orientations.
My idea to prove this was to use induction on $d$, as then hopefully we'd just need to check the case where two flags differ by their facet. This result is very simple for $d=1$. But even in the $d=2$ case I get stuck. If two flags differ by their edge, then it's very visually obvious they define oppositely oriented triangles, but I can't find a way to formalize this.