Dihedral sign patterns in ${\mathcal{M}}_{0,n}$

The connected components of ${\mathcal{M}}_{0,n}(\mathbb{R})$ are in bijection with the $(n-1)!\mathrm{/}2$ dihedral orderings of $[n]$. They are all isomorphic. We construct monomial maps between them, and use these maps to prove a conjecture of Arkani-Hamed, He, and Lam in the case of ${\mathcal{M}}_{0,n}$. Namely, we provide a bijection between connected components and sign patterns that are consistent with the extended $u$-relations for the dihedral embedding.