 Angular exchange for elastic collisions

Write incoming relative velocity as $r\omega$ and outgoing relative velocity as $r\sigma$. For isotropic scattering, the product measure $dC{r}^{2}drd\omega d\sigma$ is invariant under exchanging the two angular variables. This gives the pre/post-collision weak identities. Keeping $\sigma$ fixed and sending only the velocities to the outgoing pair is not invertible: that map discards the incoming angular direction.