I don’t recognize a Dodgers-Giants “rivalry”. The last time both teams were concurrent contenders was the 1950s and maybe the early ’60s. In my lifetime, the Dodgers have had runs of success in the late ’70 and early ’80s — times when the Giants sucked out loud. The ’90s featured the “Brian Johnson Game”, but the teams of the ’90s were the Yankees and the Braves. The Dodgers’ last playoff appearance was in 2008 (2009?) another year when the Giants were horrible.
L.A. is Laker Town, and the Dodgers stopped being a relevant franchise even before the current owners bankrupted it. As far as the Giants go, given their bizarre predilection to tank against crap teams, look to Giants-Nats if you want to see a rivalry.