The group 2H_5 has ten distinct (oriented) reflections x -> x' = rxr, rr=1.
Each reflection is characterized by a 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.
r = a, aba, a(ba)(ba)=-b(ab)(ab) ... blue
r= -a, -aba ... light blue
r= b, bab, b(ab)(ab)=-a(ba)(ba) ... light green
r= -b, -bab ... green.
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 pentagon. 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_5.
Remark: Instead of r we could also use the dark red vector s = r I2 (where I2 is the [positively] oriented pseudoscalar of the plane) to describe the reflections instead by x-> x'=-sxs = rxr. s is perpendicular to the line of reflection. This description seems indeed to have been chosen by D. Hestenes in his paper "Point Groups and Space Groups in Geometric Algebra".
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).