  Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping
class group, and $\mathcal{T}(S)$ its Teichm\"uller space with the
Teichm\"uller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and
consider the subset of $\mathcal{T}(S)$ fixed by $H$, $\mathrm{Fix}(H) \subset
\mathcal{T}(S)$. For any $R>0$, we prove that the set of points whose
$H$-orbits have diameter bounded by $R$, $\mathrm{Fix}_R^T(H)$, lives in a
bounded neighborhood of $\mathrm{Fix}(H)$. As an application, we show that the
orbit of any point $X \in \mathcal{T}(S)$ under the action of a finite order
mapping class has a fixed coarse barycenter. By contrast, we show that
$\mathrm{Fix}^T_R(H)$ need not be quasiconvex with an explicit family of
examples.
