Everything BB above has said is true. Head-Fi has a way of amplifying the differences--often minute--between different pieces of gear and different audio formats. This is probably to be expected since it's an enthusiast forum for personal audio, but nonetheless HF culture tends to project the notion that if it ain't summit-fi and doesn't come in 24/192, it ain't any good.
Personally, I haven't experienced this to be the truth at all. As BB says, it's all about how well you're satisfied with the gear you have, and that's as much a mentality as it is a function of how well the gear itself actually performs. I currently have in my modest stable a former flagship (Beyerdynamic DT880) and an underappreciated monitoring model (Shure SRH440) which have so far survived any intentions I've ever had to upgrade or replace one or the other. Every time I consider upgrading, I ask myself, "What specific, fundamental improvement will this upgrade afford over what I've already got?" and the answer is invariably, "There is none." My gear does everything I need it to do, and I'm not convinced (despite being a long-time Head-Fi member!) that anything I could realistically buy (or, really, anything in existence) would be a quantum leap in performance over what I already have.
That's not to say that my gear is perfect, but then, there's no such thing as a perfect piece of gear. I happen to know that I can live with the flaws my current gear has, and that I can thoroughly enjoy the music through it, which isn't something I can necessarily say about whatever I might get as a replacement, even if it costs $$$$.
When it comes to compression formats, after having read so much about how lossy compression destroys music, I didn't trust my unaided judgment whether or not I could hear a difference at higher bitrates. I instead worked out where my own personal transparency points are via ABX testing. The results were enlightening, humbling, and liberating all at once. I imagine it's theoretically possible that somebody could consistently split 320 mbps MP3 from the original source file, but I happen to know that I cannot (I top out at LAME's V0 VBR setting, which is ~245 kbps), which is all that's important for my own listening.
The upshot of what I've said in the previous paragraphs is, when I slap my headphones on, the only thing I worry about is which performance of the Brandenburg Concertos I want to listen to at that moment. It's a great feeling not worrying that somewhere out there, there might be a headphone that theoretically inches a minute distance further toward transparency than what I'm currently listening to, or that somewhere in the file I'm listening to, hundreds of kilobits were thrown away by the encoder.
All of this said, I'll offer some general suggestions for what to look for in a good headphone for classical. Generally, you want a neutral tonality with good extension in both directions (i.e. a tonality that doesn't focus unduly on any one region of the sonic spectrum and which produces usable response through the lowest and highest frequencies audible to human hearing). For comfort and fidelity reasons, you'll want a circumaural (over the ear) design. You'll also likely want an open back design as opposed to a closed back one, unless isolation from your surroundings is of paramount importance. Theoretically, it's not physically impossible to produce an exemplary closed-back headphone, but in practice they're not as common, mainly due to engineering difficulties involved in extracting a natural response from a driver in an enclosed cup.
Looking at your OP, the models you've selected are excellent choices for all of the above. The HD650 is going to be more laid back and fuller bodied than the HD800, but not to excess. In fact, plenty of people enjoy their HD650s for classical to this day, even though newer, shinier models are available. You might also look at the HD600, which is a little closer to neutral than the HD650. There's an excellent InnerFidelity article about the HD6x0 line here. I can also vouch for my DT880, having had many years experience with it and having enjoyed it thoroughly for classical, though to be honest, there probably isn't a tremendous amount it can do that an HD800 wouldn't also accomplish (albeit for a lot more money).