The faux boombox contains those totemic five albums, all remastered by Jones, plus three bonus CDs, containing singles and rarities and unheard early demos. Alongside this is memorabilia like badges, posters, stickers, a DVD with previously unseen footage shot by Don Letts and Julien Temple, two reprints of the band fanzine Armagideon Times as well as a new issue edited by Simonon, containing rousing memoirs.
On the box, it says "Made by the Clash" in big letters; it really feels as if the band have invested a lot of their time. Simonon has talked of "claiming back our Clash heritage". Four years ago, however, Jones told me that he responded to the public's spiralling interest by trying not to think about it - curating that heritage has weighed heavily upon them all.
"This box set is about a responsibility to our legacy," says Headon, their drummer from April 1977 onwards. Back in 1982, it was Headon's drug addiction, which sparked the band's collapse. He spent the ensuing 20 years as a junkie and alcoholic, until he sobered up nearly 10 years ago. Though he appeared in 2002's documentary about the band, Westway to the World - "I looked like a corpse", he admits - he has largely been a silent partner, until now.
Headon first entered Jones's orbit in late 1975, passing an audition for the flamboyant guitarist's pre-punk group, the London SS. "Mick had long hair, and he was wearing a white Afghan coat," he recalls. Headon, schooled in jazz, soon left for better things, and only crossed paths with Jones again by chance, outside a Kinks gig at the Finsbury Park Rainbow in 1977. By then Jones' crew were punk-rock heroes, but desperately needed a new drummer.
"We'd come along a bit," Jones smirks, "so when I asked, he was much more like, Oh right, I might actually join these guys."
In the interim, Jones' attitudes towards music (and fashion) had been changed by seeing the Sex Pistols in 1976. With Simonon on bass, and legendary provocateur Bernie Rhodes as manager, they set about poaching Strummer from his pub-rock outfit, the 101'ers. Rhodes door-stepped Strummer, offering him the frontman role, and 48 hours to decide.
"But then, Bernie calls him," Simonon remembers. "Okay, I've changed my mind from 48 hours to 24 - what's your answer?"
Such coercive tactics might easily have dissuaded Strummer, but he relished the opportunity to radicalise Jones' group. On Sound System, an early demo captures one classic song mid-transformation: "London's burning with boredom, baby," runs the chorus - the "baby", which smacked of old-school rock parlance, was later changed to a more urgent "now".
However, it was after their inflammatory first album, Jones believes, "that it all clicked - once we got Topper in". With Headon's powerful, dexterous drumming as its motor, each successive Clash release expanded beyond four-square punk into reggae, hard rock, jazz, funk, gospel, disco and early hip-hop. The thunderingly portentous anthem London Calling - an update of Edward R Murrow's radio broadcasts for the nuclear age - fronted up a double album of genre-defying diversity and depth - their masterpiece.
Selfless and sleep-deprived, the Clash hurtled from tour bus to recording studio to airport, without respite for six years. "We built up such momentum," says Jones, "the intensity was incredible. We were fed up with each other, we'd been together all the time. We should've just had a holiday."
"But that," reasons Headon, "wouldn't have been the Clash, would it?"
When Strummer called a band meeting in 1982 and fired the drummer for his druggy unreliability, Headon and Jones cried. Strummer soon brought his differences with Jones to a head, and gave him the bullet too.
The Clash's self-destruction on the cusp of conquering America rescued them, in the words of their roadie, "Baker", from a descent into "pantomime and derision". Jones, Simonon and Headon concede, though, that all concerned took decades to get over their sense of loss.
Simonon, now 57, talks of Strummer agonising beneath "a cloud of disappointment, that what was great had been fractured by sackings and Brutus-style assassinations on the steps". While Jones enjoyed success with Big Audio Dynamite, and Simonon became a painter, both joining Damon Albarn's Gorillaz band in 2010, it was Headon who was hit the hardest. Today, by comparison with his ghostly appearance 10 years ago, he is unrecognisable - healthy and measured. "I wouldn't change a lot of it," he says. "I've had a fantastically varied life: I've been in the Clash, I've been in prison, I've been in hostels for the homeless, I've busked on the Underground, I've lost my spleen in a car crash, and was in intensive care for a week. I was a nervous wreck, terrified of my own shadow.
"Now I'm extremely happy. Being involved with Mick and Paul again is a dream come true. It's like Joe's been with us, too, through his words. It's as close as we can get, without reforming."
As early as 1986, Strummer, Jones and Simonon were reconciled, but they always refused lucrative offers to reunite and are unlikely to do so. For one thing, Headon, now 58, doesn't drum anymore because it is too physically demanding. Putting together the box set, however, has been therapeutic.
"It was just an opportunity to do something now," says 58-year-old Jones, "to represent the music. I like the idea of the ideas carrying on somehow - like Che Guevara's. It's also a restoration, because the tapes would've rotted soon. So it's just the recorded works, presented as best possible - no different from the complete series of Kojak or Breaking Bad." He grins. "That's how everyone buys things these days, isn't it?"
But do they not see the Rolling Stones, still together in their 70s, and wish the Clash had overcome their conflicts, and soldiered on?
"Different band, different generation," states Jones, himself a committed Stones fan. "In the end, it couldn't have been any other way with us, or it wouldn't have been the same story."
That story, as told in the words, music and iconography of Sound System, remains one of rock's most inspirational yarns.
Who: The surviving members of The Clash
What: Sound System box set and new best-of Hits Back
When: Out now
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- TimeOut