Can I Charge for Assault and Battery?

I live in NYC and on the subway, this guy is playing his music aloud right next to me and a few others (we are all seated). I tell him to stop playing the disturbing music, he refuses, and a verbal altercation ensues. We insult each other, he calling me “pussy” and I calling him “uncivilized trash". His stop comes, and before he leaves the car, he spits at me. I am held back by others on the train, but eventually I follow after him up the stairs, finding him at the upper platform. We both exit the turnstile and soon after I shove him. He punches me, I fall down, and while on the ground I grab hold of his leg. The details afterward are unclear. He punches me either two or three times afterwards. I might have fell to my knees and then he punched me for the second time, or perhaps I fell straight to the ground and the punches after the first were all while I was on the ground. I am sure that afterwards when I was on the floor and holding on to his legs, he punched me at least twice, one of which was definitely to the back of my head. People nearby in the station immediately separate us, and in the midst of this I throw my only punch at him, which is powerless, and I don’t recall if it connects.

Afterwards, I talked to the police and they told me that both of us would get arrested. He mentions the following charges but doesn’t to specify for which of us it is: “provoking, stalking, assault.”

My physical situation from the incident is as follows. I was bleeding profusely from my right nose for a long time, and it is diagnosed as a nasal fracture. My left cheek is severely swollen. There is a swelling bump on the back of my head now, and I am not sure if it’s from falling on the back of my or because of his punch.

My main questions are as follows. What legal action can I take against the man? (I am thinking battery for spitting, assault for punching me over a shove, which I believe is not self-defense, especially for pummeling my head even after I was on the ground) In turn, what legal action can be taken against me? (Provoking? Stalking?) Ultimately, with these two in mind, is it a good idea for me to attempt to take legal action?

A secondary concern I have is this. The officer mentioned that I can be arrested too if I wanted to take legal action, and they couldn’t find the man. If I want to call the police now and create a report or case of the incident or to press charges, then, even if the man isn’t pressing charges against me, could I still be arrested or charged?

What motivates me to call the police now is mainly out of concern for the future. If I see the man again in the future, I can potentially press charges against him. Or, if he harasses me or does anything, having this case documented now will be helpful for then.