I think it varies. I'll use Iowa as and example.
Iowa/Minn, Flyod, a bet was made between the Gov. of Iowa, & the Gov of Minn for a award winning pig. Iowa has been winning Floyd pretty constant lately so even though it is for a trophy, It doesn't seem like much of a rivalry.
Iowa/Wisc, Heartland, started in 2004(iirc), current record is split evenly something like 41-40-2. so this is pretty competitive but it feels (to me) like a trophy made because the serious is so even.
Iowa/ISU, CyHawk, well, this trophy is, imo, a trophy of the middle of Iowa were there are "split" families. even though it makes sense to have a trophy between them it feels really forced.
Iowa/PSU, none, PSU doesn't consider this a rivalry game because they have such a great history. Iowa doesn't really consider this a rivalry game b/c we've been kicking their ***.
Iowa/NW, none, NW fans see this as a rivalry game b/c they've had our number in recent history. Iowa doesn't see it as a rivalry b/c.. well seriously. it is NW.
Iowa/MSU, none, this game has been pretty competitive over the last 15 years I think, the home team has won every time but once. Last year. Tends to be a hard hitting game with emotion and he who makes the least mistakes usually wins.
Iowa/OSU/UM, none, would love to get to the point where these are considered rivalries but we don't win them enough.
Iowa/Everyone else, none, when both teams are good there will be some emotion for a rivalry game, but when either 1 or the other or both aren't, nope.
Now as for
Iowa/Nebraska, none (yet), I truly expect to see this become like like the Iowa/Minnesota rivalry. bad blood and border war. but I'd be fine with it being like the Iowa/PSU rivalry, where you have your history but we still can kick your ***.