  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 $\bar M_{0,n+2}(R)$.
