From my perspective, it's the Commanders. The Commanders have stopped us from going to the Super Bowl twice.
And they always seem to trip us up or beat us when they're not expected to. Who can forget the Santana Moss comeback or the fluky defeat during our 13-3 year or the loss with Colt McCoy that may have stopped us from securing home-field advantage throughout the playoffs in 2014? And, remember, we kept the Commanders from an undefeated season in 1991.
For the most part, the Eagles rivalry really started with Buddy Ryan. Yes, they stopped us from going to a Super Bowl also, but they haven't really beaten us in meaningful games or when they've beaten us it's because we really sucked as a team.