Construct a symmetry group for the set of integers on the number line that generalizes the dihedral group to have a countably infinite, rather than finite size. Treat the integers as vertices.
What exactly would you need to construct this symmetry group? What properties other than the flip and the rotation would you need to show? I know that the flip would just be multiplication by $-1$ and that rotation counter-clockwise would be adding $1$ and rotation clockwise would be subtracting $1$.
When we generalize a concept from finite set to infinite set there could be many possible directions for generalizations. Your concept of rotation is plausible.
There are reflections in finite cases swapping two elements and keeping the others fixed. In the group generated by the two elements you describe there does not seem to be any simple swap. If such an element is desirable you should go for a different definition.