They do not commute. Draw a square, label the vertices, and try it out. Dihedral groups are not abelian.
Rotations and reflections do not generally commute.
For example, consider reflection in the x-axis in R^2, and counterclockwise rotation by 90 degrees. Then the vector (0,1) ends up in (1,0) upon reflection and then rotation, while it ends up in (-1,0) upon rotation and then reflection.