According to Nielsen-Thurston classification, given a closed surface $S$, the elements of the mapping class group $\mathrm{MCG}(S)$ lie in three categories: periodic, reducible and pseudo-Anosov.
Can the type of an element of $\mathrm{MCG}(S)$ be recognized from its centralizer?
For example, it is known that the centralizer of a pseudo-Anosov element is virtually cyclic. Can the centralizer of a reducible element be virtually cyclic as well?
Here is a finite order mapping class whose centralizer is infinite cyclic. Think of rotating a 3-tined eggbeater by angle $2\pi/3$ (here's a picture of the 4-tined eggbeater, the best I could find quickly). Consider a surface $S$ of genus 2 cut into two pants $P_1,P_2$ by three nonseparating curves $c_1,c_2,c_3$. Let $f : S \to S$ be a finite order mapping class that permutes $c_1,c_2,c_3$ cyclically, preserving each of $P_1,P_2$. The centralizer of $f$ is virtually generated by the multi-twist which twists once around each of $c_1,c_2,c_3$.
On the other hand, the virtual centralizer of $\phi \in \text{MCG}(S)$ does detect pseudo-Anosov mapping classes. This is the subgroup $VC(\phi)$ of all $\psi \in \text{MCG}(S)$ which commute with some nonzero power $\phi^i$. The group $VC(\phi)$ contains an abelian group of rank $k$ where $k$ is sum of the maximal number of curves of a reducing system for a power of $\phi$, plus the number of pseudo-Anosov component surfaces of the Thurston decomposition of $\phi$; and that value of $k$ is maximal. So $k=1$ if and only if $\phi$ is pseudo-Anosov.