Here's something pretty great via Craig Calcaterra of NBC Hardball Talk: In a court case, the Minnesota Twins are designated as the arch-rivals of the White Sox.
"Fricano asserts that the Cleveland Indians are the arch rival of the Chicago White Sox. While the two teamsmaintain a healthy rivalry, this court notes that it is generally accepted, at least among informed baseball followers,that the title of arch rival belongs to the reviled Minnesota Twins, to be shared, during inter-league play, with theChicago Cubs."
That's from a 2004 court case over a brawl in the stands at U.S. Cellular Field. So, in the court of law, the Twins are the main rivals of the White Sox.
I do like the language that says the Cubs share the role of arch-rivals but only during interleague play. That's assuming some White Sox fans don't root against the Cubs during league play, which is a pretty false assumption to make.
The whole rival thing is interesting with the White Sox -- back in the late 90's and early 2000s, the main rival to the Sox was Cleveland. But with the rise of Minnesota in the last decade, the Twins vaulted the Indians in that regard.
There's no long-standing rivalry between the White Sox and another team like Dodgers-Giants, Cubs-Cardinals or Red Sox-Yankees. So the Sox cycle through rivals -- in a few years, maybe it'll be the Royals.
But yeah, right now, the White Sox and Twins are arch-rivals. You probably didn't need a court of law to tell you that, though.