  We study two families of quantum models which have been used previously to
investigate the effect of topological symmetries in one-dimensional correlated
matter. Various striking similarities are observed between certain
$\mathbf{Z}_n$ quantum clock models, spin chains generalizing the Ising model,
and chains of non-Abelian anyons constructed from the $so(n)_2$ fusion category
for odd $n$, both subject to periodic boundary conditions. In spite of the
differences between these two types of quantum chains, e.g.\ their Hilbert
spaces being spanned by tensor products of local spin states or fusion paths of
anyons, the symmetries of the lattice models are shown to be closely related.
Furthermore, under a suitable mapping between the parameters describing the
interaction between spins and anyons the respective Hamiltonians share part of
their energy spectrum (although their degeneracies may differ). This spin-anyon
correspondence can be extended by fine-tuning of the coupling constants leading
to exactly solvable models. We show that the algebraic structures underlying
the integrability of the clock models and the anyon chain are the same. For
$n=3,5,7$ we perform an extensive finite size study -- both numerical and based
on the exact solution -- of these models to map out their ground state phase
diagram and to identify the effective field theories describing their low
energy behaviour. We observe that the continuum limit at the integrable points
can be described by rational conformal field theories with extended symmetry
algebras which can be related to the discrete ones of the lattice models.
