To make the idea concrete, one could do plane geometry with the dual quaternions. One could use dual quaternions to represent rotations and reflections in the plane. A point object can thus be represented as a 180-degree rotation about a point, and a line objects can be represented as a reflection about a line. There's therefore no need to think in terms of bivectors.
Framing it in terms of vectors/bivectors is what allows us to quantify these ideas. (i.e. have to write down your planes/lines/points with numbers at some point).
But you got the essence! The abstract/coordinate free/geometric/group theory ways of thinking about this are most insightful.
So in PGA, vectors are reflections, other isometries are combinations of reflections, and the geometry are the associated invariants. (including planes, lines, points and screw axi).
A fun extension is to replace the reflection by an inversion (reflection in a sphere). Vectors now become inversions (leaving spheres invariant), the rotors become the conformal group (composition of two inversions), leaving circles invariant, etc .. the corresponding Clifford Algebra is CGA (R(4,1), with a proper parametrisation).
