Separation coordinates, moduli spaces and Stasheff polytopes

We show that the orthogonal separation coordinates on the sphere $S^n$ are naturally parametrised by the real version of the Deligne-Mumford-Knudsen moduli space $\bar M_{0,n+2}(R)$ of stable curves of genus zero with $n+2$ marked points. We use the combinatorics of Stasheff polytopes tessellating $\bar M_{0,n+2}(R)$ to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates and St\"ackel systems from the mosaic operad structure on $\bar M_{0,n+2}(R)$.
Comment: Extended version, 20 pages, 4 figures