Lyr 2 versus Asgard 2
Conclusion first. The Lyr 2 is a better amp and I prefer the Asgard 2 anyway (and it is the one I am keeping). The Lyr 2 presents a more detailed, nuanced, and precise sound and image. Overall, the sound is more punctuated (some might refer to this as fast). If you listen to large, complex symphonic recordings exclusively, there is no doubt you should go with the Lyr 2. On the other hand the Asgard 2 is warmer (not in temperature, they are both really hot, and the Lyr 2 is the hotter of the two), and bigger sounding. I have a feeling this Euphonia is a biproduct of higher second order distortion in the output. Normally this is the type of sound that is applied to tube amps, but in this case the solid state was way more "tubey" than the tube amp (I didn't do and rolling though, and have very little desire to do so). In the end this largeness helped the HE560 by filling out the low end (if not actually giving more impact), by providing a little warmth and soundstage. The HE560 is already plenty detailed, more than enough to make many of your recordings sound sub-par. The Asgard 2 is somewhat compensatory then. It is a silly word, but musicality and my ability to emotionally connect with the music is improved with this combination. That said, I think I liked the Lyr 2 slightly more with my AKG Q701s (which doesn't need help with the sound stage or sounding big...with the exception of the low end). It is hard in some ways to let go of the Lyr 2, it might sound great with other headphones if I don't stick with the HE560. On the same hand big dark headphones might not sound as great on the Asgard 2. But the reality is that even though there are differences between the Lyr and Asgard, they are not enormous or really anything more than subtle. Given that I like the sound better with the he560s and the Asgard 2 is half the price of the Lyr 2, keeping the Asgard doesn't feel like much of a long term risk anyway.
Note: I ended up doing most of my testing on low gain, but switched over to high gain for some comparison. Everything I observed seems to apply to both gain settings, but for whatever reason (I have some theories) is less pronounced with the high gain setting.