  Moduli spaces of points on $n$-spheres carry natural actions of braid groups.
For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of
mapping class groups of positive genus surfaces, by establishing exceptional
isomorphisms with certain moduli of local systems. This relies on the existence
of group structure for spheres in these dimensions. We also use the connection
to demonstrate that the space of rank 4 Stokes matrices with fixed Coxeter
invariant of nonzero discriminant contains only finitely many integral braid
group orbits.
