I have a wonderful set of Sennheiser HD650's that I adore.
Many will prefer HD600s
But at $180, you could look as Senns HD598.
I prefer OPEN BACKED headphones for classical music.
Closed are perhaps better for bass heavy dance type music.
Also you'd probably do best looking for more Neutral cans for classical. (Senn's included, many cans are bass oriented.)
If you can find a pretty lightly used set of HD600s, get them, but the HD598 should be significantly more comfy to listen to than the AT-M50X.. AKGs are pretty good too, but I still prefer Senn. I have owned many over 20 years and loved them all. Even my earbud CX-985 are amazing.
A point to remember:
Very high end cans require good amplification and a long play-in time. My 650s needed almost 150 hours of playing before opening properly. But they are 300ohm. The 598's will be easier.