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).