For me the Art Ensemble of Chicago fall into the classical slot even if I buy their work from 'Jazz'
https://www.youtube.com/watch?v=3PMcRuh8O74
Dave Brubeck was fresh out of classical training when he produced his best work.
https://www.youtube.com/watch?v=MFbOE5GuiBE
Some of the great practitioners of Jazz like Gillespie, Bird and Monk were very aware of classical composers like Schoenberg and Bartok.
https://www.youtube.com/watch?v=eeM0JMgj358
On the other side there are pieces like Milhaud's Le Boeuf sur la Toit and Pierrot Lunaire by Schoenberg that presage developments in Jazz, but most like Shostakovich and Stravinsky failed entire to grasp it.
Despite some of the really great musicians of all time being Jazz people overall there are rigid constraints in mainstream Jazz that do not exist in the Classical tradition. My father was a semi-pro jazz pianist and I'm trying to remember what they were. Something about not moving from the rhythmic and harmonic base of the original theme?