Conference teams often have them

• They may be between schools
• What schools in the same district have, often
• Siblings may engage in them

Wiktionary Word definitions in Wiktionary
n. (plural of rivalry English)

The Collaborative International Dictionary Word definitions in The Collaborative International Dictionary
Rivalry \Ri"val*ry\, n.; pl. Rivalries . The act of rivaling, or the state of being a rival; a competition. ``Keen contention and eager rivalries.'' --Jeffrey. Syn: Emulation; competition. See Emulation .