I thought Dodgers Giants was the west coast equivalent.
There's no Babe Ruth component with many, many decades of being cursed, or not quite making it. Also, the Giants and Dodgers have often had their peaks at different times. Sure, there's a rivalry, but it's not the internationally known one of Red Sox v. Yankees. I don't think there can ever be a baseball rivalry quite like those two AL East titans. We like to think at times it's great and for us here, it's as big as it gets, since I don't see the same animosity of A's against the Giants, or Padres against either LA team, or norcal teams against socal teams.
I don't know if Giants and Dodgers games make news on the east coast. But over here, the whole Boston v. NY thing is still big news, especially after 2004.
The big sports rivalry on the west coast is definitely college related more than anything else, not a professional sports rivalry. That would be any UCLA team against any USC team, and up north mostly Stanford football vs. Cal football. The sports pages in California get more mileage from these non professional rivalries. But I did wish there was some professional sports rivalry of the same magnitude every year outside of the sometimes close race in the NL west. If ever there is a year when the NL west leader spot is between LA and SF and both teams have a good chance of going to the WS, year after year, then I can see some news happening outside of the area of a true rivalry worth mentioning to the rest of the country. Whenever I hear of a Red Sox team and Yankees team making the postseason, it's also not unrealistic to think one of those teams will eventually walk off with a WS ring.