As a pretty heavy classical listener, I can say the HD 600 works quite well. I've also owned the DT880 for a little over seven years now, and I would say classical is one of its strong suits. There are three major differences between the two:
1) The DT880 has better bass extension than the HD 600, which rolls off pretty noticeably below around 70 Hz. For classical I've found this is really only an issue for things like organ bass and large percussion instruments. Most of the time I barely miss the lower bass on the HD 600, but the DT880's extra extension is certainly nice.
2) The HD 600 has a bit more forward midrange than the DT880. The DT880 is probably closer to neutral here, but this is somewhat tempered by point three. And this is not to say the HD 600 is inferior in the midrange; on the contrary, I prefer its presentation here to the DT880 most of the time (again, because of point three). The DT880 tends toward clear, open midrange with a bit more perceived distance from the source than with the HD 600, whose midrange is a bit more present and upfront, though no less clear.
3) The DT880 has noticeably boosted treble compared to the HD 600, which has what I would consider optimal treble balance. In particular, the DT880 has more presence in the upper treble, which emphasizes the sense of space and adds a clear, shimmering quality to strings. I particularly like this effect, and it's the source of the "airy" descriptions of the DT880 sound, as well as the feeling that it has a larger soundstage than the HD 600. On the other hand, the boosted mid-treble affects the timbre somewhat and makes the overall sound seem a bit thinner than the HD 600—the latter is slightly euphonic, whereas the DT880 leans towards analytical. I prefer euphonic to analytical these days, so the HD 600 is an overall closer fit for my sonic preferences.
The treble is both the DT880's most characteristic sonic feature and its biggest flaw, as it's the area where the sound deviates most strongly from neutral. This is less of an issue for classical than, say, modern pop music, as with the latter you often have hyped mid-treble baked into the mastering, which leads to uncomfortable sibilance on the DT880. For classical, even with vocals, this has not been an issue IME.
Hopefully this helps. I like both headphones for different reasons, but I spend most of my time listening to the HD 600. Where it falls behind the DT880, it isn't very far behind, whereas the DT880's treble can be more troublesome, often being recording dependent. The HD 600 is closer to neutral overall and disappears for me in a way the DT880 can't quite match.