I've owned the HE-560 extensively and had the chance to demo the HE-X. I would say that the sound signatures and tonal characteristics are quite different between the two models. I would personally prefer the HE-X over the HE-560, but when factoring in price point and value, it is a harder call. I know that some folks are bothered by the lower treble peak of the HE-560. The HE-X did not have an issue in that region from my initial impressions. Would need more time with them to really get a more in-depth view of their sound signature. My impressions left me with the feeling that the HE-X did have extremely well-done bass that has plenty of warmth and heft without bleeding into the midrange. Seems more sonically similar to the HE-1000 rather than the HE-560 in overall tonal balance. I personally have difficulty hearing the splashiness of the treble some folks mention with the HE-1k, so I am not a good judge of whether that exists with the HE-X.
Overall, for its current price point, I do see it as an extremely competitive option against the LCD-X, LCD-3, and Ether Open/Closed. I personally would lean towards the HE-X personally with the Ethers next followed by the LCD-X. If shopping for a sub-$2k pair of headphones. The other two new sub-$2k headphones that are coming out that I am personally interested in are the Dharma and HD800S. The HE-X and Dharmas would have been the most impressive new open headphones I've heard with the Ether Closed being a very well-done closed pair of headphones for shoppers looking for a closed-back that does not sound as closed as most other options on the market.
HE-560 I would say is more variable to personal preference than the rest of these options, but I did find the HE-560 to compete relatively well against the HD800 with some subtle difference sonic characteristics that can make you choose one over the other depending on the price point. The HD800 has subtly larger left to right sound stage and more pronounced mid-bass and upper treble in relative comparison to the HE-560. The HE-560 has better overall bass in my personal opinion with more linear extension and more sub-bass presence. The HE-560 has more of a lower treble emphasis. Both of these options are a bit more on the brighter side of neutral in my personal estimation. The HD800 can be quite peaky in its treble region, so finding a good amplifier pairing is essential. The ALO Studio Six was an extremely impressive pairing for the HD800. I personally do also prefer a slightly warm tube amplifier with the HE-560. Not something as warm like the WA7 which I feel like diminishes sound of its soundstage and imaging detail, but the Lyr 2 worked extremely well for the HE-560. I feel like the HE-560 can work better with mid-tier amplifiers, while the HD800 may lead people into investing in much more expensive amplifier components to really get it to sound just right.
Just some of my personal impressions & thoughts. YMMV.
