Detective Da Fino, the PI hired to find Bunny Lebowski and played by Jon Polito, expresses his admiration of The Dude's apparent playing one side against the other throughout the movie. In Miller's Crossing, Jon Polito plays a mob boss who is victim to and absolutely hates a double-cross.
Walter's oft-repeated line "shut the fuck up Donny!" is precisely written because Buscemi's character wouldn't shut up in Fargo.
Replicas of The Dude's Medina Sod bowling shirt and the Big Lebowski's "Person of the Year" mirror can easily be found for sale.
Time Magazine even made a mirrored cover of their own for Person of the Year in 2006.
The below-mentioned "Dudeism" is a recognized religion, and people have been married under it.
Fake Nationality: The German nihilists played by Peter Stormare (Swedish), Flea (Australian), and Aimee Mann (American). The fourth nihilist was played by an actual German, Torsten Voges.
Parody Religion: The film has spawned its own, The Church of the Latter-Day Dude, or Dudeism.
Playing Against Type: To some degree, John Goodman as Walter Sobchak. John Goodman's characters, when not directed by the Coen Brothers, tend to be nice, huggable jolly guys. Walter is anything but.
Which nearly became his type. Goodman said in an interview that a lot of the roles offer to him post-Lebowski were variations of the blustering Walter character.
Also, Steve Buscemi's role as a clueless, laid-back slacker who can't get a word in edgewise was written as a jokey inversion of the motor-mouthed psychopath types he'd played in Fargo and Reservoir Dogs. Walter's oft-repeated line "shut the fuck up Donny!" is precisely because Buscemi's wouldn't shut up in Fargo.
The Red Stapler: The White Russian has become somewhat legitimized as a "guy" drink thanks to this movie, as most people nowadays will just assume a man ordering one is a fan of The Dude.