Well, the "natural" rivalry between all Dy[i]namo teams is not quite the one: every time before or after games in Riga, Minsk or Moscow fans of the respective teams meet in big groups and have laughs in the same sport bars, they also play hockey versus each other. It's a very friendly "rivalry".
So I would say that just as in other leagues the regional rivalries are bigger. The Moscow triangle (Spartak, CSKA, Dynamo Msc), the green derby (Ufa vs Kazan) and the battle for Chelyabinsk Oblast (Traktor vs Magnitka). I think Lev and Slovan might also develop one.
Nothing is close to Omsk vs Vityaz. Those teams HATE each other. I mean besides the numerous cheap shots, the bench clearing brawls, the most penalized game in hockey history, Vityaz's ineptness probably killed Cherepanov.