My favorite Gen VI Pokemon is Pyroar, and during my runthrough, I pretty much exclusively used my Pyroar with Rivalry as my lead. Anyway, so I beat the game a few days ago, and I went to check all my Pokemon's IVs I had captured to see if I got anything good.

Turns out my Pyroar has max Sp. Def IV. And that's not all, when I checked a lot of my boxed Pokemon which I had caught along the way, out of around 3 boxes or so, 8 of them all had max Sp. Def IVs. These weren't Baby Pokemon or Friend Safari Pokemon, they were ones I had encountered just playing the game.

So, I went out and caught more Pokemon with Pyroar in the lead, and it turns out, I found about 3 Max Sp. Def IV Pokemon from a batch of 30; none of them had max IVs in anything else bar one with Max HP. I tried catching Friend Safari Pokemon as well, and that turned out even better, as 3 of about 20 Pokemon had 3 Perfect IVs including Max Sp. Def.

It seems Rivalry somehow makes it more likely for you to encounter Pokemon that share your highest IV I'm guessing.