I've played piano for over 15 years, so I have a decent idea of what they're supposed to sound like.
The HE-560 does well with classical piano. It brings out the sweetness in pianos that tend that way. Bass is of course excellent - you get the feel for the richness in the lowest octaves of a large concert grand - something that is missing in many headphones. The low mid-range is just a hair too forward; but this is a problem that the majority of headphones suffer from. Staging is good. For me, many headphones make piano sound like the music is just emanating out of the air; with the HE-560, you get a sense of being seated a couple rows back in a moderately sized hall.
By way of comparison I like it better than the DT 880 and much, much, MUCH better than the T1 gen 2. The T1 made piano sound like ****. I note that you already have the HD 650; I haven't heard it, so I'm not sure if the HE-560 will be better or not. My view is that you should be prepared for a side-grade. Some things may be better with the HE-560, other less so.
On that subject, if I had to point out one flaw of the HE-560, it would be the slightly uneven upper mid-range. The area from 1-3kHz is just a bit too recessed, while 4-8kHz is just a bit too forward. Not hugely noticable, and not a big deal, though. Just an area where other headphones perform a bit better.