  Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable
surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and
sufficient conditions for two finite-order mapping classes to have commuting
conjugates in $\text{Mod}(S_g)$. As an application of this result, we show that
any finite-order mapping class, whose corresponding orbifold is not a sphere,
has a conjugate that lifts under any finite-sheeted cover of $S_g$.
Furthermore, we show that any torsion element in the centralizer of an
irreducible finite order mapping class is of order at most $2$. We also obtain
conditions for the primitivity of a finite-order mapping class. Finally, we
describe a procedure for determining the explicit hyperbolic structures that
realize two-generator finite abelian groups of $\text{Mod}(S_g)$ as isometry
groups.
