Photons can interact with matter, particles, nuclei, etc.
But can they interact with one another?
By interaction I mean any kind of physical interaction: Momentum/Energy transfer, electromagnetic field superpositions, polarization, etc.
If so, can this interaction be described through both a particle and wave behaviour by the photon?
Photons can interact with each other. However, the interaction is very weak in the vacuum. I'm not sure this can be explained using a simple explanation from wave-particle duality. The full explanation would come from Quantum Field Theory. Heuristically you could think of this process as both photons splitting into virtual positron-electron pairs, those particles interacting, and then the pairs annihilating to give two more photons.
Wikipedia has a good article on this here
Most of the time they interact by changing a medium which changes the properties that the photon sees.
But another possibility is HOM interference, in which the possible outcomes of the different photon outcomes interfere quantum mechanically.
