In my opinion, the DT 880 and the HD600 are both excellent choices. In my opinion, the DT 880 is slightly more unforgiving with bad recordings (including some Led Zeppelin songs), and the HD600 sounds slightly better overall, but I really like both with classic rock (and other sub-genres of rock and metal).
^ This sums it up well! Just grab a magni (or vali for a warmer tube-ish sound) and a modi, and you'll be set. I enjoyed classical more on the DT 880 but Metal was a bit better on the HD600 IMO. You can't go wrong with either. Overall I would give the edge to the HD600 for what you need, but if saving a bit of cash or simply *enjoying classical a bit more than the other genres* (IMO*) is your deal, then the DT880 won't disappoint one bit. They're both incredible, I couldn't imagine you being disappointed in either! Good luck!