When electromagnetic waves of any nature pass or meet one another I believe there is the potential for constructive and destructive interference. While this is technically not an "interaction" between waves, which @Ben Crowell pointed out has a specific meaning with respect to electromagnetic waves, I assumed you used the term "interaction" as generally defined in the dictionary being "mutual or reciprocal action or influence" (Webster). If that's not what you meant, please let me know.
Once they pass they generally resume their original form in the absence of any dispersive medium. I would think that the degree of potential interference would depend on how similar the wavelengths are, the closer they are the greater the degree. To that extent, I wouldn't think microwave and x-rays, far apart in wavelength, would involve much if any interference.
That being said, I think of greater interest is how different types of electromagnetic waves interact with matter, particularly regarding ionizing vs non ionizing radiation and human tissue. For a good overview on the interaction of electromagnetic radiation and matter, I recommend you check out the following link:
Hope this helps.