= Solution
Rotations commute with one another, while reflection conjugation sends $R^j$ to $R^{-j}$. Thus the rotational <conjugacy classes> are pairs $\{R^j,R^{-j}\}$, except when the two entries coincide. For reflections, direct use of the relations gives
$$
R^s(R^jm)R^{-s}=R^{j+2s}m,\qquad m(R^jm)m^{-1}=R^{-j}m.
$$
If $n$ is odd, multiplication by two permutes <residues> modulo $n$, so all reflections are conjugate. The complete class list is
$$
\boxed{\{I\};\quad\{R^j,R^{-j}\}\ (1\leq j\leq(n-1)/2);\quad\{R^jm:0\leq j<n\}.}
$$
Their sizes are $1$, then $(n-1)/2$ classes of size two, then one class of size $n$, giving $(n+3)/2$ classes and $2n$ elements in total.
If $n$ is even, the half-turn $R^{n/2}$ is central and forms its own singleton. Conjugation changes a reflection exponent by an even amount or negates it, so it preserves parity and reaches every exponent of that parity. The list is
$$
\boxed{\begin{gathered}\{I\},\quad\{R^{n/2}\};\qquad\{R^j,R^{-j}\}\ (1\leq j<n/2);\\
\{R^{2j}m:0\leq j<n/2\},\qquad\{R^{2j+1}m:0\leq j<n/2\}.
\end{gathered}}
$$
There are $n/2+3$ classes; their sizes are $1,1$, then $n/2-1$ pairs, then two classes of size $n/2$. Geometrically the two reflection classes correspond to vertex-axis and edge-axis reflections, in either order depending on the choice of $m$. These are the <conjugacy classes of a finite dihedral group>.
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