I read a good portion of Harpo Marxs biography. Groucho would spend a lot of time reading books as a kid. He was very good on stage, but it seemed very much enjoyed spending time alone.
For this reason I think he is probably an introvert, that is if we assume that a type remains fixed. If not, he still could be one who has learned to interact socially.
Harpo Speaks! is a terrific book and from what I've heard and read the most factual of the brothers' books regarding what went on in the pre-film days. Monkey Business by Simon Louvish is another great one and for things you can't find anywhere else the out-of-print and hugely expensive "Marx Brothers Bio-Bibliography" is worth getting your hands on (via library). I love it!
Groucho was a bookish solitary child and later on in life needed time alone to write his books in the late 20s and 30s. Because of the time period/generation he did attend a lot of parties so he was no recluse but yeah, I have a hard time buying him as an E. Cusp, at least.
He entertained but often his nasty side would emerge. Harpo didn't cover it in his book but his wife Susan does say that Groucho would intentionally make his wives cry if a party was hosted at their place. Perhaps this was some kind of anxiety thing?
To add onto this, it was usually always mentioned before Groucho overtook Harpo in popularity*, that while the other brothers were a hit with the ladies Groucho wasn't that much of a hit and hooking up didn't come easy to him.
*usually the best and most accurate accounts of Groucho before his self-architectured mythos took over.