There exist a number of established rivalries between larger cities, and I am curious about how they have developed and are maintained.
First a delineation: I don't mean the traditional national anti-Capital sentiment, nor the arrogance of a larger city vis-à-vis a smaller suburb. I also don't mean a rivalry that is only based on sports (the Schalke-Dortmund type for those of you acquainted with German soccer), though sports of course is one of the mechanisms to maintain such a rivalry.
Let's start with a few German examples:
1. Run for No.2: Hamburg-Munich (of course enhanced by religion, geographic and cultural distance, etc.)
2. Economic vs. political regional centre: Leipzig-Dresden, Lübeck-Kiel
3. Historical discrimination (that's an interesting pattern I only came across after some research): Cologne-Düsseldorf. Cologne used to have the trade monopoly on the Rhine, meaning trade ships by-passed nearby Düsseldorf. When industrialisation set in, however, Düsseldorf took over. Moreover, when Prussia gained possession of the Rhineland, they selected Düsseldorf as provincial capital, deeply hurting Cologne's pride (Roman founding, largest German city throughout the middle-ages).
Don't forget Karlsruhe/Mannheim (category 1) and Bruchsal/Schwetzingen!