Versor, Pin and Spin

A versor is a product of invertible vectors in a Clifford algebra . The versors are exactly the Clifford group , where is the twisted conjugation given below. That the versor group coincides with the Clifford group is a classical theorem, and versors are the algebraic carriers of orthogonal transformations.

Definition of twisted conjugation (the versor action)

The action is , with the grade involution. On an even versor , so is a rotation. On an odd versor , so is a reflection. The involution is what makes send regardless of parity, hence realises orthogonal maps by sandwiching.

Pin and Spin cover the orthogonal groups

Normalising the norm gives the subgroups . Here , defined by , is a 2:1 cover of , and is a 2:1 cover of . The scalar factor cancels in the action, since , so .

Conjugation by a versor is a co-isometry

With respect to the Clifford scalar product , the adjoint of the inner automorphism is (using and ). Hence for , where is scalar, . Sandwiching by a versor therefore preserves the scalar product, which is what makes versor-valued transport orthogonal.

Equivariance, reachability, and why chirality survives

Read the twisted action , with the grade involution, as a group action on a feature space. A layer is then -equivariant when it intertwines all of , and merely -equivariant when it intertwines only the even subgroup. Since whereas , these are different demands, because -equivariance constrains behaviour under reflections while -equivariance does not. Equivariance is moreover distinct from reachability, which orthogonal maps a versor product can actually realise. The two are frequently conflated, yet a layer can be Spin-equivariant while its representation still carries reflection-sensitive grades.
That distinction is the lever for handedness. An -equivariant intermediate representation keeps every grade, including the top-grade pseudoscalar that separates from an ordinary scalar by the sign it assigns to odd elements. A scalar readout collapses to grade and destroys that bit, which is the mirror-image degeneracy behind enantiomer discrimination. Keeping live in the pipeline, so that pseudoscalars stay distinguishable from scalars, is what makes a versor-based encoder chirality-sensitive.