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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 $2$. 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Dhanwani","submitted_at":"2019-01-31T11:49:48Z","abstract_excerpt":"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 $2$. 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