What's the outcome of two polarized electrons beam collide head on? Let's say that I have two electron beams targeted towards each other along x-axis. Both beams are polarized on axis (z-axis) perpendicular to x-axis. Another beam's polarization is up and another down on z-axis.
What would happen when those polarized electrons collide (with low energy)? Just plain elastic collisions? Has somebody ran such an experiment? Any references?
 A: For transversely polarized electrons incident on unpolarized charged particles, the magnetic moments of the polarized electrons will see the oncoming charge as a current and be steered by it. You get an asymmetry in the scattering that's perpendicular to the direction of the spin.

Here's a cartoony way to think about it. A dipole-field interaction evolves to minimize the free energy, which is basically proportional to the volume of strong field. Point the electron's magnetic moment up. If the electron passes to the left of the nucleus, the induced field points down, antiparallel; the dipole can reduce the volume of strong field by cancelling out the stronger field nearer the nucleus. If the electron passes to the right of the nucleus, the induced field points up, parallel; the dipole can reduce the volume of strong field by combining with the weaker field farther from the nucleus. Either way the electron gets steered to the right.
This effect for electrons scattering from nuclei is called Mott scattering and was actually the original evidence that the electron has a magnetic moment way back in the 1920s. Interestingly the first measurement of Mott scattering discovered that beta-decay electrons are longitudinally polarized, an effect which no one understood at the time and which was basically forgotten until after the discovery of parity nonconservation in 1957.
The polarization-induced asymmetry in Mott scattering is due to the nuclear charges, I suppose, because the nuclei are heavy; if an electron and a nucleus exchange some amount of momentum, the electron gets deflected more.
If you collided two polarized beams, you'd have a second effect due to the dipole-dipole interaction. I think that this would predominately show up as a torque on the dipoles: the polarizations of the scattered beams would be rotated slightly compared to the polarizations of the incident beams. That would be a hard experiment to measure a small and uncontroversial electromagnetic effect. Storage-ring colliders generally have unpolarized beams, because the spin direction precesses in the field of the steering magnets. Electrons which have gone around the ring ten times will have a different polarization direction than those who have gone around the ring twenty times, and you would have to work very hard to tell one bunch from the other in a collision, and then you'd have to analyze the polarization of the secondary beam. It's not impossible, though. An exquisite measurement of muon spin precession in a storage ring is the heart of the muon $g-2$ experiment.
anna v mentions the planned measurement of parity violation in Møller scattering at Jefferson Lab and correctly points out that it's not quite what you're asking about: they have a longitudinally polarized electron beam on a fixed target of unpolarized hydrogen. However, the beam polarization isn't completely longitudinal, and the parity-allowed asymmetry from Mott scattering is much larger than the parity-violating asymmetry due to the weak interaction. Sometime during the running of that experiment there'll be a week where the accelerator is configured to produce mostly transverse polarization, both to measure what its effect is in the detectors and to quantify the magnitude and direction of the leftover transverse polarization during longitudinal running.
Your prompt reminds me that scattering of the (longitudinally) polarized electron beam from the transverse-polarized electrons in an iron foil is a standard technique for measuring beam polarization at Jefferson Lab – it's called Møller polarimetry and there are working Møller polarimeters in two of the four halls. You'd need to do some work to decide what you want to learn from transverse-transverse scattering that can't be learned from the standard longitudinal-transverse setup, at a level that would satisfy the laboratory's program committee. It'd be a challenging measurement: the polarimeters as they're built now are designed to be blind to transverse polarization. (In fact, during the transverse polarization measurement for another parity experiment a couple of years ago, we used the Møller polarimeter to confirm that the beam polarization was completely transverse, by watching for its asymmetry to disappear.) And the program committee would push back on scheduling you, even if you did impress them: transverse running at JLab changes the polarization to all the halls, and involves a couple of days of downtime on either side.
A: Electron electron scattering is called "Møller scattering" and has been calculated to higher order for the needs of an experiment that must be in preparation or running about now. 
The experiment  is with a high energy beam of polarized electrons with the objective of measuring the weak mixing angle

The MOLLER experiment at Jefferson Laboratory proposes to measure the parity-violating asymmetry in electron-electron (Møller) scattering. The measurement would be carried out by rapidly flipping the longitudinal polarization of electrons that have been accelerated to 11 GeV at CEBAF, and observing the resulting fractional difference in the probability of these electrons scattering off atomic electrons in a liquid hydrogen target

The electrons in hydrogen are not polarized and the beam is polarized in the direction of flight  so it is not the setup of your problem. 
I suppose the authors of the paper would be able to answer the question for low energy vertically polarized beam beam scattering.
My expectation would be elastic scattering, or an X-ray produced in the field of each other. 
A: My question, concerning protons, got its answer from this document (Chapter 3), Anomalous $A_{NN}$ Spin-Spin correlation. 
