I was trying to think of the best interstate rivalries (sorry OU/TX, OSU/UM) from a competitive atheltic program perspective nationwide. And not including the obscure sports, but the main ones — football, basketball (M/W), baseball/softball.
I thought of in no particular order:
Clemson/South Carolina
Kentucky/Louisville
Iowa/Iowa State
Alabama/Auburn
Utah/BYU
If you included gymnastics, skiing, and volleyball — Utah/BYU could be one of the best rivalries from a pure competitive standpoint across multiple sports in the nation.