Well, that's exactly what they are:
Sol is Order. It is rigid and unmoving.
Luna is Chaos. It is fluid and evolving.
Too much of either is bad. In the right amounts and mixed well, it's awesome.
Anywho, to go into more detail... the first thing we might want to do is create a world. Worry first about the small stuff - what will be the new names? What would our governments be like? How will we run the nations? What are the nation's borders?
After that, it's much easier. For example, I usually see Jupiter as being Imperialistic. What's the first thing that comes to mind when you think Empire? The evil empire from stuff like star wars. That's a serious image problem that we would have to face, and a lot of the conflict would be the extreme amount of distrust that those in, say, a Republic might have against us. Wars of ideals, debates, and culture. That's how the political events would be - intellectually driven, in heated arguments and flared tempers.
Then you have oneupsmanship. These are competitions like Collosso. Tests of strength, fighting off monsters and always trying to do better than the other nations, and of course, duals to prove who's the best at combat. But nobody said that this is strictly about strength - there could easily be two divisions. The Warrior Division, and the Caster Division - where it's the power of their spells that win out. This is a post-Golden Sun world, after all. Alchemy is well known and common.
Then you have the underhanded competitions. These would be done in mostly a roleplay kind of fashion, but they're also probably the trickiest to do, probably with restrictions to ensure fair play. For example, one vs one, with a judge from another clan to ensure no godmoding. The point of these? Let's say there's some documents. You know, stuff with intel. And they need to sneak in and steal them. Well, the trick is this - where do you hide them, where do post guards, etc? This could be done in a kind of labyrinth/minotaur style, and heck, there might even be actual flash games or something out there we could literally use to play it, just with people typing out their moves from the game into the forum to make a fun story out of it.
And, of course, we have casual competitions, where a representative from each clan is chosen to play stuff online such as Board Game, Cards Against Humanity, and other fun party games.
The point is to do more than just fight.