Things were going so well for the Rolling Stones in 1981 that even an album of leftover tracks dating back to their occasionally underwhelming mid-'70s records was heralded as a big event. That album of outtakes, Tattoo You, even surpassed expectations, hitting No. 1, yielding their last chart-topping single and becoming arguably the last great record of their still-thriving career.

But it's not just the reworked Goats Head Soup, It's Only Rock 'n' Roll and Black and Blue cast-offs that made Tattoo You a keeper; it was also the Stones' renewed energy and creative rebirth following that so-so trio of albums (which really had no chance of stacking up to the exalted run started with 1968's Beggars Banquet and continued through 1972's career-peaking Exile on Main St.). Some Girls, the 1978 LP that got them out of the rut, went as far as to borrow inspiration and sounds from the music that was making the Rolling Stones obsolete as the decade neared its end, including punk and disco.

All of those mixed '70s emotions about the group – from "The Greatest Rock 'n' Roll Band in the World" to lazy caricatures of themselves – were filtered into Tattoo You, which carried the Stones into the '80s with their crown still somewhat in place. A four-CD box set celebrating the album's 40th anniversary confirms fans weren't just riding a post-Some Girls high: Today, the record sounds just as committed and airtight, and maybe just a tad scraped together.

The original 11-track LP, remastered here, serves as the center point of the box. "Start Me Up" and "Waiting on a Friend" were the big singles and remain the highlights, but buried cuts like the slow-burning Black and Blue leftover "Worried About You" find new relevance in the Stones catalog. Rediscovery makes the disc essential, but it's the other three CDs, featuring studio rarities and a live show, that bring together the era for a more complete portrait.

Nine tracks earmarked during the sessions but eventually abandoned were recently completed by the band, freshened up with new vocals by Mick Jagger and Keith Richards' guitars. Only one of the songs – an early reggae-lite version of "Start Me Up" – found its way on the 1981 album, leaving covers of the Ch-Lites' "Trouble's A-Comin," Jimmy Reed's "Shame, Shame, Shame" and Dobie Gray's "Drift Away" on the cutting-room floor. Resurrected now, these songs would probably struggle for a place on Tattoo You, though the power-riffing "Living in the Heart of Love" most deserves a spot, recalling classic-era Rolling Stones in its directness. (Other originals "It's a Lie" and "Come to the Ball" also crackle with the bluesy stomp recovered during the Some Girls era.)

Sharing its name with a 1982 live album by the Stones, a complete concert from Wembley Stadium found on the third and fourth discs called "Still Life" comes off like a victory lap rather than an indispensable tour document. Stage-toughened – or stage-wearied, depending on your perspective – versions of "Under My Thumb," "You Can't Always Get What You Want," "Brown Sugar" and "(I Can't Get No) Satisfaction" often betray Tattoo You's hurried and slapdash approach. The Rolling Stones were working at such a pace that records and tours were stitched together with eyes on the clock.

That an album like Tattoo You came out of this period is no small feat. This 40th Anniversary Edition doesn't shed much new light on the project; the record's story had already been properly told. But it does give a more clear look at what could have been (more covers, maybe fewer Black and Blue scraps) and the LP's impact not even a year later – especially "Start Me Up," which instantly received a prime spot in the Stones' set lists. Four decades on, the album still matters.

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