### Reflections in the symmetry group 2H_2 of a symmetric bar (dumbbell)

The group 2H_2 has four distinct (oriented) reflections x -> x' = rxr, rr=1. Each reflection is characterized by the unit direction vector r of the line of reflection. These reflections are oriented, because each line of reflection has two possible opposite directions, characterized by the sign of the unit direction vector r.
We have:
r = a
r = -a
r = b
r = -b

Each reflection can be visualized by dragging the red vector perpendicular to the line of reflection to a particular value of r. It is easily verified that any other choice of r, not in the above list, will not produce a reflection symmetry of the symmetric bar. The combination of two such reflections at r and r' will yield an (oriented) rotation y->R~yR=(r'r)y(rr') from the dicyclic group 2C_2.

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