  A theorem of Ritt states the a linearizer of a holomorphic function at a
repelling fixed point is periodic only if the holomorphic map is conjugate to a
power of $z$, a Chebyshev polynomial or a Latt\`es map. The converse, except
for some exceptions, is also true. In this paper, we prove the analogous
statement in the setting of strongly automorphic quasiregular mappings and
uniformly quasiregular mappings in $\mathbb{R}^n$. Along the way, we
characterize the possible automorphy groups that can arise via crystallographic
orbifolds and a use of the Poincar\'e conjecture. We further give a
classification of the behaviour of uniformly quasiregular mappings on their
Julia set when the Julia set is a quasisphere, quasidisk or all of
$\mathbb{R}^n$ and the Julia set coincides with the set of conical points.
Finally, we prove an analogue of the Denjoy-Wolff Theorem for uniformly
quasiregular mappings in $\mathbb{B}^3$, the first such generalization of the
Denjoy-Wolff Theorem where there is no guarantee of non-expansiveness with
respect to a metric.
