Like Wilco’s Yankee Hotel Foxtrot, Fiona Apple’s Extraordinary Machine comes with a juicy backstory about the big, bad recording industry. Wilco, as everyone knows, pleaded artistic integrity when its label charged that the band’s recordings weren’t sufficiently commercial; bolstered by tons of press, Internet buzz, and a big-screen documentary-cum-hagiography, Wilco prevailed. When YHF was eventually released in its original form by a division of the very label that spurned it, the CD became Wilco’s biggest success, both commercially and critically. Apple’s story has a murkier moral. When her label refused to approve the tracks that she’d recorded with longtime producer Jon Brion, the singer, who hadn’t released an album in six years, fell into one of her famous funks and simply gave up. It wasn’t until an anonymous insider leaked the recordings on the Internet that the hype machine kicked into gear. The bootleg CD garnered gushing reviews, and Apple’s fans demanded its release on the Web site freefiona.com while deluging Sony’s offices with foam apples. But instead of pulling a Wilco and sticking it to the Man, Apple claimed that she wasn’t any happier with the Brion sessions than Sony was. She dutifully returned to the studio with Dr. Dre’s main man, Mike Elizondo, and started over. One of the many charms of Apple’s second album, the magnificent When the Pawn . . ., was the contrast between Brion’s hyperdecorative production style and Apple’s decidedly visceral approach: Surrounded by the elaborate whimsy of Brion’s arrangements, Apple seemed more dangerous, more alien somehow, like a scorpion in a Fabergé egg. Music snobs who dismissed her on the basis of the triple-platinum Tidal, the embarrassingly sleazy video that accompanied her first hit single, the onstage meltdowns, and the incoherent rant at the MTV Video Music Awards were compelled, after Brion’s massive cred injection, to give her a second chance, enough time to notice the venomous sweetness of her voice, her jazz-singer phrasing, her ingenious melodic variations, and her savagely elegant piano playing. The masses, alas, were underwhelmed: When the Pawn . . . sold only a third as many copies as its predecessor. In its current form, Extraordinary Machine includes nine re-recordings by Elizondo and co-producer Brian Kehew, one new track, and two recordings from Brion’s original production, which serve as aural bookends. Compared with its previous incarnation, the album sounds somewhat more commercial, but the differences are less pronounced than one might expect. The arrangements still boast baroque, Brionesque flourishes — woodwinds, string sections, tack pianos, pump organs, Optigans, chamberlins, marimbas — but they’re heavier and more hypnotic, less cluttered and crazy-sounding. If Brion’s production was ecstasy, Elizondo’s is opium. Whereas Brion played almost all of the instruments besides piano, Elizondo enlisted a crack ensemble of session musicians, including the inestimable Abe Laboriel Jr. and the Roots’ Ahmir “?uestlove” Thompson on drums, whose sinuous rhythms impart a groove that cunningly underscores Apple’s soulful voice. Check out the lurching, lovelorn “O’ Sailor” and the louche, logorrheic “Not About Love,” in which Apple’s quicksilver tempo shifts seem not only intuitive but downright funky. You could dither endlessly about the producers’ relative merits, but you’d be missing the point: No matter who twiddles the knobs, it’s Fiona’s show. For proof, look no further than the searing ballad “Parting Gift,” which requires nothing more than Apple’s scorched-sugar contralto and her alternately fierce and delicate piano accompaniment, wherein lazy arpeggios give way to great clomping bass-clef chords and a bittersweet blues evolves into a molar-rattling requiem. All by herself, and with or without her label’s imprimatur, Apple is weirder than Wilco will ever be.