I'm an ENFJ, so my opposite is ISTP. I'll do my best not to generalize. I can only go on what I've personally experienced.
So, you've figured your opposite type. What you think of them?
I've always gotten along with them really well (the guys at least, as I've never met a female ISTP, to my knowledge?). I'm never sure what it is exactly that I do or say that just clicks with them (or vice versa) but we wind up grouping together and getting into trouble. Most of the ISTPs I wind up around are punks like me. Usually what kicks the door open is my clothes or a remark about the Clash or something and then it's on like gangbusters.

I've been told by a few that they find me weirdly sane and an open escape from their way of thinking when they want it. We've gone to art museums and debated over the art, and pointed at the same piece and gotten something entirely different from it, which is fun, especially when you can understand the other person's differing perspective. They've helped me like abstractions more. We laugh at the same stuff. We both value intelligence and competence and the pleasantly unexpected.
Do you feel that type is shrouded in mystery?
No.
What strikes as the oddest thing in them?
I would certainly love to know what it is that makes them seem so fearless.
Can you begin to understand what their life is about?
I won't say I know what it's like in their heads, but I will say they aren't a mystery to me. I get what they're after. But I will assert here that ISTPs in general are pretty complicated people and a lot of what goes on remains submerged so you're only seeing what they're mostly consciously allowing you to see (with occasional departures into showing what they can't help, which is reserved for friends they're sure won't betray them...). I think part of what makes an ISTP what they are is the ability and choice to change or take a different path when the impulse strikes. Impeding that would be very very bad. Betraying them (or what they take for betrayal) is a huge mistake as they don't seem to have a defense against it from their loved ones. You can really hurt them.
Do you think it's a more or less commonly understood type than yours?
I think most of the world sees ISTPs as bad mofos. There's more to them than that. But yes, I think they're better understood than ENFJs because, well, just look at James Bond/Daniel Craig. Russell Crowe. Val Kilmer. I'd say the type is well-represented.
What kind of societies or groups these people participate in?
The ones I've known have liked to haunt political arenas, car clubs, hockey venues, realms of higher learning where mastery or virtuosity was the key.
Do you see those types a lot?
I certainly used to.
How do you act around them?
Generally? Very at ease. Weirdly at ease, in fact. They don't scare me or intimidate me or make me feel like I have to act a certain way. I *do* get the impression that I'm being gnawed at sometimes, like a shark testing a surf board to see if it's food or just fiberglass, but I've never failed that particular test before, so I just let them poke and prod and squeeze the Charmin until they're sure I'm what I say I am. If the ISTP and I are mutually attracted to each other, that leads into another arena. I've been chased around a kitchen island by one at a party and it's
serious, dude. lol That's when ISTPs (and ISFPs for that matter) seem to have no brakes whatsoever and they aren't worried about it.

If they like you, THEY LIKE YOU. I stayed one step ahead of him all night so I'd leave the party with my pants on.
My ISTP best friend, no matter what was going on or who I was dating, ALWAYS had the seat next to me. That was HIS SEAT. Anyone taking his seat got a blistering look from me. He had his spot by me and that's because he never forced his opinions on me or treated me like a burden. He's one of two men in my entire life who's ever seen me without make-up on. I trusted him that much. He was the type to get up in the middle of the night and drive an a hour out of his way if I needed him, and never act like it was a problem. It truly wasn't a problem. He wanted to be there for me. He really kept me sane in some ugly ugly times. And I looked after him too.
What would you think they think of you?
I wouldn't ask. I tend to think I'm the weak link there. I don't think an ISTP would be the type to bother with someone they didn't like, so I guess that's something? They certainly seem to "get" me pretty much right away. It's eerie. I've been told that I'm humanizing, and one told me that I was comforting to have around because I could be trusted and because I cared about people so much. The ISTPs that I wind up with have also expressed a real affinity for my edginess and my particular brand of humor or curiosity about everything, but I dunno -- I've never asked one what they thought of me. They'd probably think that was a daft question to ask.