Let me get this out of the way: I know nothing about the psychology balance between Lakers and Clippers fans. I don't know if I've interacted with a representative of either camp for more than a few minutes in years, and I'm not going to try to do so now.
I'm not going to look anything up, either, because sports fans are sports fans, and I see no way the power dynamic in Los Angeles doesn't fit a specific mold repeated in other places around the country like Alabama, Michigan, and East Rutherford.
What follows is all blind presumption, and if that fact upsets even one Lakers fan, then this caveat was worth including.
You should root for the Clippers in the NBA Playoffs because they share a city with the Lakers and because a year's worth of bragging rights for Clippers fans will stir up phenomenal behavior by Lakers fans. I say this, again, without actually knowing any Lakers fans, but I'm very confident I'm right, based on knowing similar fans in other sports.
Here's what would happen every day until next NBA season if the Clippers won the NBA Finals.
In response to Clips fans boasting about it being a new day in L.A., Lakers fans would get all #RINGZ, even more so than they surely already are. Sixteen to one! Come back and talk when you win three more hands' worth! Actually, this is already happening, and I say that, again, without actually looking to see whether it's actually happening. Just trust me.
With absolute ignorance as to what it is that Lakers fans and Clippers fans actually discuss in real life, if they can even legally interact, I also assure you we'd be promised this:
The balance of power in Los Angeles will soon correct itself.
Kobe Bryant will remember how to defend, the Dwight Howard thing will fix itself one way or the other ... and I don't really know any other Lakers players without looking them up, which I absolutely refuse to do here. Does Gary Payton still play there? Either way, we'll be reminded that the Lakers will be infested with trademark greatness, literally greatness that they have trademarked, again next year and the filthy, lowly Clippers will be right back to tanning rat leather in their underground shanties of squalor and despair, even though the Clippers seem like they'll probably be pretty good for a while. (I base this on their record, which is pretty good. I know nothing else about the Clippers beyond that and their two famous players.)
Despite all that huffing, we'd also be told the Lakers don't consider the Clippers a rival. Nor the Spurs or Kings or Suns or Warriors. Only the Celtics are worthy. Am I correct in this assumption? I do not doubt that I am. This would be said so many times, you'd start to think the Lakers and Clippers are actually really big rivals. This bluster, also, is probably already happening, based on what I consider to be likely without actually investigating.
We'll also see the most fascinating thing of all: Lakers fans, who cheer for one of the four most successful pro sports franchises in America, accusing Clippers fans of bandwagonery and front-running. Again, I assume this is already happening and refuse to actually go find examples of it happening, but I assure you, if it isn't happening yet, it will.
The point is that you should root for the Clippers to win because it will make Lakers fans react in hilarious ways.