According to the question Do positively charged particles exchange photons? there was an answer
Yes. Photons are the carriers for the electromagnetic force, regardless of the charges involved.
followed by a comment
Unless of course the charge is zero ;)
Now I'm getting curious how neutrons get scattered of each other?
The neutron is not an elementary particle. Its overall charge is zero, but depending on the energy available the fact that it is composed out of charged quarks becomes important. Thus both the strong force and the electromagnetic will play a role depending on the energies of the scatters and the targets.
For low energies , like the ones coming from fission reactions in reactors, matter is practically transparent to the neutrons due to their zero charge, and in order to absorb and contain them a lot of matter is needed. They can interact with the strong force with the nuclei of the atoms, be captured etc, generating secondary radiation centers.
Due to the quark structure the neutron has a magnetic moment that can interact with magnetic fields.
To interpret correctly the crossections the quark nature has to be taken into account where also the electromagnetic interactions will enter, example here., mathematical models are necessary.
In condensed matter research, neutron scattering experiments are very useful to study the magnetic structure of materials.
Neutron do indeed possess a magnetic moment, and thus interact with the local magnetic field. Electrons have a much larger magnetic moment, yet neutrons are used precisely because they have zero charge and do not interact electrically, but purely magnetically.
In the end it is still an EM interaction though. Microscopically, it is one of the charged quarks that interacts with the EM field.
At higher energies (read: particle accelerators), neutrons can also interact through the strong nuclear force.
Neutrons scatter from other neutrons and protons through the nuclear force or "strong force", which is also what keeps neutrons within the atomic nucleus. The strong force is also what holds the constituents of nucleons (neutrons and protons) and other hadrons (the whole spectrum of mesons, baryons) together. The nuclear force which binds can also cause them to scatter. If you look at the diagram on Wikipedia, note that there's a very large repulsive force when the neutrons are too close.
Because of the repulsive part of the nuclear force, a neutron with a large kinetic energy will thus approach too close and bounce off another nucleon, whereas a slow neutron might bind with another nucleon.
Neutrons, although electrically neutral, consist of electrically charged particles (quarks) bound together as a neutral whole, and so they also have a magnetic moment.
(Aside: The references used in that Wikipedia article are all very obscure books. I was looking for something to read about this but was disappointed. I'll have another search on this site and then look it up again.)
