Compared to Apple dongle on PC. On the other hand, Apple dongle is far from DX1 in detail.
DX1 is transparent, what's more it runs on an AKM chip and people who believe in chips having sound will all tell you AKM has warm, smooth and full sound, while Cirrus Logic (Apple dongle) usually is cold and crisp
Anyway, the first assumption should be that Apple one is wrong (but we have it measured and we know it is decent aside one issue), whether some kind of low end distortion, something related to Cirrus Logic hump (the mentioned issue), or simply underpowering the headphones. However I've ran into an issue with Apple dongle where it would change the sound depending on USB host, although in my case it was boosting treble and sounding thin (so more like high frequency distortion...). While desktop DACs theoretically should be more resilient against influence over USB, I do recall seeing someone mention he picked up interference on DX1 while another USB powered DAC was fine. So if there is any issue with host affecting the DACs it could affect both Apple and DX1. The simple way to check that would be to use a different PC or even a mobile phone, although Android has its own ways of affecting sound so you should only compare both DACs on the same platform (also try an app with exclusive USB mode like free HiBy Music or paid UAPP, you need to set it in settings and close the app with exit button to restart first though).
But honestly, the answer you will find the most is probably - the difference is in power. Followed by question "Have you level matched the loudness/volume on both?". Not matching the listening levels exactly is the number 1 issue causing differences in perceived sound where there shouldn't be any. Moreover, to my annoyance I haven't tried HD560S yet, but with most Sennheisers of this construction they kind of like being ran loud, the top end and details come up to the point when listening on lower levels might even feel muffled for some in comparison. This almost exactly matches description of your observations. US Apple has 1Vrms on tap, EU/Global Apple has just 0.5Vrms, DX1 has 3.89Vrms, that is a considerable difference, especially since HD560S was measured by DIY Audio Heaven (user solderdude found here) to have 108dB/V sensitivity.
As such I would simply suggest using Equalizer to adjust the sound to your liking. There is simply a good chance that whatever other good desktop DAC you would buy instead, the result will be the same as with DX1. But by all means, if you want to get to the bottom of this, try perfect level matching the signal out of the devices, and different hosts to figure out if there is an inherent difference between the sources or not.
