The Euclidean group consists of three transformations: translations (runs), rotations (twirls), and reflections (flips) and their concatenations. Corresponding to the first two transformations, we have, respectively, linear momentum and angular momentum. But nothing corresponding to the third. Howkum?

Cartan's Theorem says that translation and rotation reduce to reflection. This should mean that linear and angular momentum reduce to reflected momentum. I call it "bimomentum". When you write a matrix for it, you get the sine-cosine of half-angles that you do for quantum spin and the spinor in Clifford Algebra (of which the simplest form is the "complex number").

Also, we note in an accompanying file that Bachman reduces all of Euclidean geometry to reflection.

Wotyagonnadobowtit?