Jasper headcanons bc I've thinking about this all day:
1) You'd think she's an extrovert based on the way she acts, but she's actually quite introverted. Social situations are overstimulating to her--esp the mushy-touchy-feely type you get from Steven and the CGs--and her immediate response is aggression.
2) The only social situation she really enjoys is sparring but most gems aren't willing or able to spar with her, so she ends a lot of pent up energy that she doesn't get to release until it's spilling over.
3)She's basically in a constant state of overstimulated and understimulated, to the point where she's always lashing out in frustration. That's why she seemed almost happy to have found a worthy opponent in Steven, as unhealthy as their dynamic was.
4) In general, Jasper has very specific needs that haven't been met and boundaries that aren't easily respected by the gems at Little Homeworld. She's been been through a war, lost her Diamond, separated from the closest beings to family for who knows how long, idolized and feared by those beneath her, (probably) hated by her equals, and pressured by her superiors. She's been poorly socialized throughout her life, to day the least.
5) If she can't have someone who can match her physically, she needs someone who will respect her boundaries. Luckily, she manages to find both in Bismuth, who matches her in strength, pushes her buttons in just the right way to make their sparring a satisfying experience, and backs off when Jasper's had enough. She responds best frank, blunt conversation and smack talk, which is Bismuth's specialty.
6) Amethyst was the only gem to really go and check on her after the whole Steven situation. She was resistant at first, but she has begun to realize how alone she really was after Steven essentially abandoned her. So, she started to accept Ame's attention and eventually regarded her as a sister. She still isn't at all affectionate, but they spend a lot of time together nonetheless.
7) Skinny and Carnelian were the absolute closest things to family Jasper ever had prior to Amethyst and they were the first gems she ever cared about, before the gems in charge filled her head with their own prejudices. She actually wasn't a piece of shit in the beginning and openly cared about them, but the war changed her. This drove a rift between them that Ame encouraged them to bridge, now that Jasper was a little more approachable. Now they're closer than ever and they tease Jasper relentlessly to keep her humble. She tolerates it because she missed them.
Okay enough of the deep shit, here's some fun headcanons:
1) Amethyst starts a club/hang out spot of sorts for quartzes and other ex-soldiers to hang, spar, and generally let loose. Sometimes (often) it gets a little too rowdy and it's difficult to break it up. She invites Jasper in to check the place or and maybe make friends. Jasper surprisingly shows up and is an amazing deterrent. One look from her is sobering enough to stop the rowdiest gems. That's how she became an unofficial bouncer.
2) She and Amethyst eventually started an underground gem wrestling tournament by accident as the new and improved "Purple Puma" and "Tiger Millionaire" (with Steven's blessing). She thought a millionaire was too weak and a philanthropist was too sappy ( once she learned what those were) and became "Tiger Shark" instead (still maintaining the business gimmick, like an investor, but she also has sweet fucking serrated teeth).
3) When the Betas heard Amethyst calling her "Sis" they immediately started calling her that, too, much to Jasper's irritation. That didn't last long, however, bc once they heard about her wrestling persona, "Tiger" was just too good a name to pass up.
4) Skinny pinches Jasper's gem nose and shakes it gently (in the same way you would pinch someone's cheek) as an affectionate gesture. She's literally the only gem allowed to do this to her, although she acts like she doesn't like it.
5) Bismuth likes Jasper but has an ongoing feud with her over Jasper's complete disregard for other's property. When asked why she does this, Jasper shrugs and punches a hole through another wall. What wacky and destructive thing will she do next? No one knows.
6) Bismuth also enlists her for some heavy work gathering resources for some building projects, along with some other heavy lifters. Freckles--or "Nice Lapis"-- is over of the gems on this team. The others, having heard vague rumors about the Malachite situation, were a bit weary of these two being around each other, but they hit it off surprisingly well. They don't hang out at all otherwise, but they like to team up on the job.