I believe they WERE meant to fight (or oppose each other), ever since the So6P split his powers into his two sons one obviously had to be yin and the other yang. The descendants of them would carry that trait, if not teaching it to the younger generation. There has always been a main yin and yang though.. at least from what we've seen.
Farthest back we've seen is the two sons.
Second farthest back we've seen is Hashi and Madara's fathers.
Third farthest back we've seen is Hashi and Madara themselves.
The Forth one is obvious with Naruto and Sasuke.
(you could include Obito in there somewhere, but he's been out to get just everyone in general and not just the yang).
So your statement of "they weren't meant to fight each other, but to make one clan" is a bit off. I'm sure that was the So6P intent (to have both sons work together) but the second you split the yin and yang up and say one is going to lead the other.. it's asking for trouble. He should have chosen both sons in the beginning, that is the main flaw in this story (well it wouldn't be a story if it wasn't like this) It's always been ONE side. Hashi was going to let Madara rule the Leaf alone.. That is why I think at the end of this manga Naruto and Sasuke will BOTH rule the Leaf, if not the Shinobi alliance.