  In this paper we consider the effect of symmetry on the rigidity of bar-joint
frameworks, spherical frameworks and point-hyperplane frameworks in
$\mathbb{R}^d$. In particular we show that, under forced or incidental
symmetry, infinitesimal rigidity for spherical frameworks with vertices in $X$
on the equator and point-hyperplane frameworks with the vertices in $X$
representing hyperplanes are equivalent. We then show, again under forced or
incidental symmetry, that infinitesimal rigidity properties under certain
symmetry groups can be paired, or clustered, under inversion on the sphere so
that infinitesimal rigidity with a given group is equivalent to infinitesimal
rigidity under a paired group. The fundamental basic example is that mirror
symmetric rigidity is equivalent to half-turn symmetric rigidity on the
2-sphere. With these results in hand we also deduce some combinatorial
consequences for the rigidity of symmetric bar-joint and point-line frameworks.
