Well technically the war would have already started because of Ulfric Stormcloak, but perhaps one of the two Princes were the ones to get him to try it in the first place, manipulating him?
Sanguine is the Deadric Prince of Debauchery the Darker Natures of Man(sin, lust, sloth, gluttony, greed), loves to play pranks, and loves to tempt mortals into sinful acts even more. I'm open to suggestions for a larger purpose behind starting/fueling the war, but it doesn't seem outlandish that he might do it simply for the shit and giggles and, well...doing what he loves most, to tempt mortals into committing sin.
And Clavicus is known to grant mortal desires in a false way, or even to simply allow some to try granting their own wishes in a way that ends in disaster for his own amusement, so he might be in on it for similar reasons and simply to cure boredom by playing the mortals like pawns in a game.
So I could probably see Sanguine working on Ulfric's side of the war while Clavicus fuels the Legion's side? Each playing one side or the other like a chess game, with each side being their pieces/pawns?(and I'm sure it would severely piss off those of the third side that become involuntarily dragged into the war because of it, which would give characters a nice, motivating personal grudge to put a stop to it XD)
Again, if you guys have ideas for a darker purpose behind their motives, I'm open to suggestions.