Sports Rivalries

In no way complete: if I'm missing anyone, let me know!

Soccer
Darmen - Many early friendlies and competitive games
Cosumar - Several lost World Cup Round of 16s. We did get an AOCAF 3rd place against them, though.
Equestrian States
Osarius - Lost an AOCAF semifinal and final to them
Kandorith, Polar Islandstates, Astograth - for being in the same Baptism of Fire
Sargossa - brothers and sisters in the Quarterfinal curse! Despite only three games between us (2-0-1 record), Sargossa are our equals in letting fans down since the 50s.
Audioslavia - The late 80s and early 90s have seen about one Audio-Ko per cycle, and usually not in insignificant matches. CAFA knockouts, WC knockouts, you name it. The band 'Audio versus Ko-oren' releases an album for every match, too.

Gridiron
Allamunnic States - Lost a World Bowl semifinal and final to them
Saintland - Lost and won a World Bowl final to them
Banija - Mostly due to the NSCF
Quebec and Shingoryeo
Gregoryisgodistan

Cricket
The Plough Islands - The Polaris Trophy
Liventia
Darmen - The Friendship Trophy and GCF Test S9's final
Krytenia - We have lost several important matches across multiple formats to them in rapid succession. That, and the triangular Anaian series.

Baseball
Banija
Newmanistan
Liventia
Cassadaigua
Zwangzug

Rugby Union
Neu Engollon - Two finals against them
Elejamie - Two finals against them

Rugby League

Ice Hockey
Gyatso-Kai
Banija - They have a team in our nation

Basketball
Saintland - Several IBC finals