  We consider the $N$-vortex problem on the sphere assuming that all vortices
have equal strength. We develop a theoretical framework to analyse solutions of
the equations of motion with prescribed symmetries. Our construction relies on
the discrete reduction of the system by twisted subgroups of the full symmetry
group that rotates and permutes the vortices. Our approach formalises and
extends ideas outlined previously by Tokieda (C. R. Acad. Sci., Paris I 333
(2001)) and Souli\`{e}re and Tokieda (J. Fluid Mech. 460 (2002)) and allows us
to prove the existence of several 1-parameter families of periodic orbits.
These families either emanate from equilibria or converge to collisions
possessing a specific symmetry. Our results are applied to show existence of
families of small nonlinear oscillations emanating from the platonic solid
equilibria.
