Linear symmetries act on the Salvetti model

For each hyperplane $H$, the nonzero functional $h_{gH}\circ g$ has kernel $H$ and is a nonzero scalar multiple of $h_H$. Thus $g$ permutes sign coordinates, possibly reversing their signs, and maps every facet bijectively to a facet. Since it is a homeomorphism of $I$, it preserves relative closures and incidence. It sends ${\mathcal{A}}_F$ to ${\mathcal{A}}_{gF}$ and sends the localized chamber $C_F$ to $(gC{)}_{gF}$.

These facts preserve both conditions defining the order of incident pairs. Consequently $g$ maps finite chains to finite chains, so induces a simplicial automorphism. Its realization permutes barycentric coordinates, continuously on each closed simplex, hence continuously in the weak topology; its inverse is induced by $g^{-1}$. Identities and compositions hold on coordinates, giving an action by homeomorphisms.

The diagonal map sends $H\times H$ to $gH\times gH$ and preserves $I\times I$, hence preserves the complement. Its inverse is the diagonal map of $g^{-1}$. Thus it is the action of [F3]. Nothing in steps 1.1, 2.1 and 3.1 constructs a comparison map between the two spaces.