  The interplay of time-reversal and $n$-fold rotation symmetries ($n=2,4,6$)
is known to bring a new class of topological crystalline insulators (TCIs)
having $n$ surface Dirac cones due to surface rotation anomaly. We show that
the proximity-induced $s$-wave superconductivity on the surface of these TCIs
yields a topological superconducting phase in which two Majorana zero modes are
bound to a vortex, and that $n$-fold rotation symmetry ($n=2,4,6$) enriches the
topological classification of a superconducting vortex from $\mathbb{Z}_2$ to
$\mathbb{Z}_2\times\mathbb{Z}_2$. Using a model of a three-dimensional
high-spin topological insulator with $s$-wave superconductivity and two-fold
rotation symmetry, we show that, with increasing chemical potential, the number
of Majorana zero modes at one end of a vortex changes as $2\to1\to0$ through
two topological vortex phase transitions. In addition, we show that additional
magnetic-mirror symmetry further enhances the topological classification to
$\mathbb{Z} \times \mathbb{Z}$
