Lyndon is also once again confusing a single photon interacting with an electron with a plane wave (a classical approximation of many photons) interacting with an electron.

I'd draw your attention in particular to equation 19 from the paper Lyndon mentioned, where we find the momentum imparted to the EM wave through the interaction. Notice that it's in the x-direction, opposite the motion of the electron, while the plane wave is initially travelling in the z-direction. That is, the interaction imparts a transverse momentum on the photons making up the wave to balance the transverse momentum imparted to the electron, so they're scattered from the original direction. Which of course is what we've been saying all along, that to impart momentum and energy to an electron, the photon can't continue to travel in the same direction.