A while back, I designed a very silly combat video game for three players which involved pools and water guns and trying to keep your balance on a float. I may actually implement it later, because it's an easy game to implement and would show off the avatar creation system I'm using - launch a tool with a sample project, basically.
The rules of the game are too stupid to go into detail about here, but in essence your techniques are limited depending on whether you're on your home float or not. If you're on your home float, you can use your hands or your foam pipe. If you're not, you can only use the foam pipe. So jumping to another player's float means you'll take them on with a disadvantage.
Most combat games are weighted in favor of offense or defense. Most one-on-one fighters such as Street Fighter try to avoid that, but most team-on-team or melee fighters are heavily weighted one way or the other. For example, in many sports it's illegal for an attacker to bump into a defender, giving an advantage to the defender. In a typical army game, creating barricades and hunkering down is typically a huge advantage when compared to charging into those barricades. In Disgaea, the upper tier combat is entirely about first strike victories, and the attackers have a massive advantage.
I think these asymmetries are really interesting, but they are most interesting when they cause a continual back-and-forth between offense and defense, and also more interesting when they are organic, generating more complexity than they contain.
For example, kabaddi is an unusual ball sport where you have to hold your breath while on the opponent's side of the field. This fills both of my criteria for interesting asymmetry.
It doesn't say "you need to give the ball to the other team after 15 seconds". Instead, it simply destroys you when you go on the offense. The longer your offensive drive, the most oxygen-deprived you become, and that takes some time to recover from even when you return to your side. There's no iron rule locking you into taking turns - instead, it allows you to weight your actions with an eye towards the longer-term consequences.
The rule also opens up a new area of specialization. You might be fast, or a good thrower, or a good catcher - or, in kabaddi, you might have excellent lung capacity. Every facet of skill that a player can have is one more way for a player to shine and for a team to be colorful together. "Doug is our primary striker, but it's Chad's role to spend most of his time lounging about in the enemy base trying to foul up their defense..."
Compare this to baseball. The rules of baseball are like iron bars, each made specifically to lock the player into a specific kind of performance. Rather than having complex tradeoffs and incentives, the sport is structured specifically to make teams go head to head with a very specific pattern. If someone does something that is within the rules but isn't part of the intended ritual, precise rules are laid out to force them to perform the ritual properly. For example, there are dozens of rules about how a pitcher can pitch. Must be within this precise rectangle. Must have these characteristics. Must be caught properly. Must be a ball in perfect condition. Must be... and so on.
This is the difference between a "restrictive" rule and a "facilitating" rule. Fundamentally, a rule prohibits or enforces a particular behavior, so in both cases something is being required or outlawed. However, a restrictive rule exists to reduce the complexity of the topology of the game, while facilitating rules increase the complexity of the topology. That is, restrictive rules give the players as a whole fewer options and paths, which facilitating rules give them more options and paths.
Holding your breath is a facilitating rule. It does require you to do something - hold your breath - but in turn it creates a whole bevy of complex behaviors that players can be good or bad at, all of which also affect their core performance at other facilitating rules (such as how you can throw the ball, how you can score, etc).
Requiring the pitcher to throw within a particular rectangle is a restrictive rule that exists solely so that the batter can continue to use the specific skill of batting. It doesn't open up new topologies, it carefully builds walls so that you can't go to those new topologies.
... Those new topologies of hitting people in the face with an 80 mph hunk of rubber.
Obviously, restrictive rules aren't bad. They're just not as interesting. I like facilitative rules.
So let me think of some.
The most common and fundamental facilitative rule is having a single ball. The act of controlling a single ball makes a game fundamentally weighted in favor of whoever is holding the ball. Additional rules about how the ball can be moved around can facilitate more complex play - such as the dribbling rules in basketball, or the no-hands rules in soccer.
But I like the idea that the ball is actually a dangerous thing. For example, in most modern capture-the-flag matches in FPS games, the flag makes you move slower, or limits your weapon use. Yes, you need it to score, but it doesn't give you the advantage. The game is weighted in favor of defense.
Well, you could pull a kabaddi. What if you weren't allowed to breath while you were holding the ball? Prioritizes passing the ball, interceptions, and so on.
What if there are specific limits on where you can go? For example, something like water polo. Except whoever has the ball has to stay submerged. Or stay outside of the pool, or something. I'm stuck on the whole "pool games" thing.
You can even make it so that the ball isn't a scoring device. For example, the game takes place in an unstable, shrinking island at the center of a pool. If all members of a team are in the water, that team is defeated. However, you can climb out of the water as long as you have the small handball. So if your team members fall into the water, you should toss them the ball, help them climb up, toss the ball to the next person, and so on...
I also like the whole "double ball" category of games, where there are two balls in play. There are a ton of options here. One option I like is having two balls in two colors to match two teams in two colors, and then change up the rules based on which ball is which.
For example, you have to hold your breath if you have either ball... but you can only score points with your own color. So there's a lot of game around whether you can keep your enemy's ball away from his team through repeatedly holding your breath and passing a ball around that is useless to you. Maybe you can focus your efforts by having one person hold both balls for a little while, allowing everyone else breath...
If you don't like the "breath holding" mechanics I'm stuck on today, you could also just use a time limit.
Anyway, it's fun to think of new mechanics. This gets much more open when you start to consider video game mechanics rather than human mechanics. For example, everyone can double jump or fly except for the guy holding the ball. Everyone can summon monsters except the guy with the ball. The ball is the source of energy, recharging all your robotic allies if they are nearby, regardless of who is holding it...