Creator Backlash: The Beasties haven't performed "Fight For Your Right" live since 1987, think it "sucks" and are annoyed that its ironic, sarcastic nature was lost on its intended target of fratboys. Also, they've apologized repeatedly for their earlier sexism and homophobia.
Executive Meddling: After the Beasties fled to Capitol, Def Jam head Russell Simmons announced that he would assemble an album created from the band's outtakes with extra contributions by others (think Tupac Shakur's and The Notorious BIG's posthumous albums). Paul's Boutique came out first, and nothing was heard about Simmons' idea again.
One of Us: See above and the below section on shoutouts in the Music tab.
The well-known example of Rhyming with Itself in "Pass The Mic": Mike D meant to rhyme "commercial" and "rehearsal", but in the take they used on the album, he accidentally repeated himself, resulting in "Everybody's rapping like it's a commercial / Acting like life is a big commercial".
What Could Have Been: In interviews, Mike D mentioned the idea of releasing a With Lyrics version of The Mix-Up as a companion to the instrumental one, with guest vocalists on every song. MIA and Jarvis Cocker were rumored to be among the contributors.