20 years out, The Dude still abides.

Jeff Bridges, John Goodman and Steve Buscemi recently reunited for the 20th anniversary of “The Big Lebowski,” the cult classic that made Joel and Ethan Coen modern legends to slackers and stoners the world over. Though “The Big Lebowski” disappointed at the box office when it premiered in 1998, earning a relatively meager $17 million against a budget of $15 million, word-of-mouth buzz and home video sales turned it into the cult sensation it is today. Thanks to The Today Show, the three cast members recently took a nostalgic sit-down through their initial reactions to the script, favorite fan interactions, and one of The Dude’s most abiding legacies.

On their initial reactions to the script, Goodman said he loved it right away. Buscemi, who plays the much-maligned Donny Kerabatsos, had the same reaction, but almost turned down the project nonetheless. “When I was reading the part of Donnie,...