The title largely sums up my question, what does happen if you either x-ray an x-ray, or point two x-ray generators at each other?
X-rays are electromagnetic waves, just as light rays are. The difference is in the wavelength (thus frequency and Energy
So your question has the same answer as "What happens if you shine light on light" or "What happens if you point a light ray at a light ray".
Classically, you will see the same effects you see with usual light rays, interference, diffraction, etc.
On a quantum-level, you will even be able to see direct interaction (light with light, or analogously x-ray with x-ray), as described in quantum electrodynamics (QED).
X-ray can interfere, that is the basis of Bragg's law. For that, however, the two x-rays have to be coherent, and for that they have to come from the same source, the source has to be small and far away, etc. etc.
People also have done different types of double-slit experiments with x-rays. Again, the x-rays have to come from the same source, and that source has to fulfill a few other criteria to be coherent. Grating interferometry is becoming a more and more popular technique.
If you point two x-ray generators at each other, then nothing happens. Photon-photon scattering can happen, but the cross section is extremely weak. To the best of my knowledge nobody has ever observed direct experimental evidence for this.