For a Dihedral group $D_{2n}$, the operation of reflection about an axis, that is in the same plane as the polygon, can be considered as a rotation in $\mathbb{R}^{3}$.
For e.g. for a square, the diagonal is the planar axis about which we rotate the square in the $z$-axis.
The commonly used rotation is about the $z$-axis passing through the center of the polygon. Then, doesn't it imply that there is no 'reflection operation'?
Of course, for the Dihedral group, we consider typically the axis passing through the vertices of the polygon rather than any edges, since the final shape orientation is preserved.
The way it's been picturized in say "Dummit & Foote, Abstract Algebra" it seems reflection is somewhat a "separate" operation. It would have been more intuitive to understand them as rotation in the plane of the polygon & rotation about the orthogonal axis, passing through the center to the plane of the polygon...
This kind of gels well if we study the symmetry of 3D or higher objects with respect to the "orthogonal to 3-D" axis passing through the center and the available axis in the dimension of the object that preserves structure. Appreciate your thoughts.