 Solution

The relation $Rm=m{R}^{-1}$ implies $m{R}^j=R^{-j}m$. Move each reflection generator to the right in any word, cancel pairs using $m^2=I$, and reduce exponents modulo $N$. Every element then has a normal form $R^j$ or $R^{j}m$, $0\le j<N$.

For the actual symmetry group of a regular polygon these are $N$ distinct rotations and $N$ distinct reflections. The determinant distinguishes rotations from reflections in the planar action. Hence the source's dihedral group $D_N$ has order $2N$, with its subscript counting polygon vertices. The calculations below use that convention.