According to Helena Bonham Carter, who is, after all, uniquely placed when it comes to observing what makes cinema’s Odd Couple tick – that’s Tim Burton and Johnny Depp – it’s a shared sensibility, an off-kilter take on life and a love of what she describes as “poo jokes”. She has made five films with Burton (and three of these with Depp, too) and does, of course, share a London home and two children with him. For the latest, Sweeney Todd: The Demon Barber of Fleet Street, she took the role as Mrs Lovett, opposite Depp as the serial killer who dispatches his victims with a slash of his cut-throat razor and a song.

This is a musical quite unlike any other: an X-rated Tim Burton horror film with tunes, buckets of blood and a leading man who owes more to Lon Chaney and The Hunchback of Notre Dame than Christopher Plummer and The Sound of Music. “It’s the perfect date movie,” giggles Burton. Burton and Depp are on their sixth collaboration. All is well in Burton’s cinematic world when Depp is its leading citizen. There are other directors who have found their muse with one particular actor – Martin Scorsese with De Niro, Ridley Scott with Russell Crowe, while the Coen brothers have taken a shine to George Clooney. But Burton and Depp have a logic all of their own, a secret, imagined universe that only they inhabit.

For Burton, Depp has vividly brought to life the ultimate gawky teenager (Edward Scissorhands), an endearingly eccentric but hopeless movie director (Ed Wood), a warped Pied Piper (Charlie and the Chocolate Factory), as well as playing Ichabod Crane in Sleepy Hollow, and voicing Victor Van Dort, whose intended is on the wrong side of mortality in the brilliant, animated Corpse Bride. These two clearly get along and bring the best out of each other, and Sweeney Todd, the long-awaited adaptation of Stephen Sondheim’s hit Broadway and West End musical, bears all the hallmarks of a classic Burton/Depp partnership.

“What makes it work? Well, it’s certainly not any deep analysis,” says Bonham Carter. “They have a shorthand and the same references and cultural frameworks. It’s a very private relationship because they are both very private individuals. It’s kind of like they don’t communicate at all, but they do, actually, exceptionally well. And a lot of it is based on a few notes from Tim about direction and inspiration and then they leave the creative process to look after itself. It’s not very pretentious, it’s not an over-intellectual relationship. I think they’re like brothers and there’s a deep respect on both sides.”

Depp first met Burton in 1989. The actor was fresh out of a hit TV show, 21 Jump Street, and Burton was casting Edward Scissorhands. “We met at a coffee shop in this hotel,” recalls Depp. “And instantly for me there was a connection and a kind of weird distance from the Hollywood way.” Burton immediately recognised an actor who was desperate to escape from the box marked “teen heart-throb” that casting directors had earmarked for him. “I always admired him for the simple reason that he always did what he wanted to do,” says Burton. “He could have gone and made millions of dollars as this great-looking leading guy. But no. There’s integrity there, there’s risk-taking in terms of making himself into different characters, and he’s got a great love of movies. He’s more like Peter Lorre or Boris Karloff or Lon Chaney than a glamorous movie star. And that’s the amazing thing.”

When you see them together, you understand what Bonham Carter means. They are from the same generation – Depp at 44 is five years younger than Burton – and both grew up soaking up the more Gothic aspects of popular culture, while feeling distanced from the American mainstream. Both now live outside the States – Burton in London and Depp in the South of France with his wife, the actress Vanessa Paradis, and their two children. Their friendship is based on a shared sense of humour and each has a tendency to finish the other’s sentences. Jamie Campbell Bower, a 19-year-old newcomer who landed a key role in Sweeney Todd, witnessed it at first hand.

“They are really welcoming, really friendly, but they do have these inside jokes with each other,” he says. “You want to laugh with them, but you don’t really know what it’s about.”

I first meet both director and actor in Venice where, the night before, Burton had been honoured with a Lifetime Achievement Award by the Film Festival, presented by Depp. At the time, Burton was putting the finishing touches to Sweeney Todd, which is now complete and already winning rave reviews – indeed, it has already been nominated for four Golden Globes in the Best Musical or Comedy category: Best Picture, Best Director, Best Actor for Depp and Best Actress for Bonham Carter. So if letting Burton loose on a Sondheim musical/horror movie – which in the States has an R rating because of the “buckets of blood” – with a reported $50 million budget was a gamble, it appears to have paid off.

“I’m not a huge musical fan, but I liked this one, I just loved it,” says Burton. “I sent Johnny the CD [of the soundtrack]. Johnny was like, ‘Great, great, great…’ and everybody was, ‘Yeah, great…’ and then it was like, ‘Um, can he sing?’ Nobody knew. I didn’t know.” But as a teenager, Depp had played and sung in a band called the Kids.

“I knew I wasn’t tone deaf because I play music, guitar and all. But I didn’t know if I was actually going to be able to sing. I wasn’t sure. So I said to Tim, ‘Let me investigate and I’ll send you something and see how you feel. And then we can talk about it.’ So I went into a friend’s studio and recorded My Friends – a song from the show – and I sent him that and he liked it.”

Sondheim himself was delighted to get Burton on board as director. He also trusted that Depp wouldn’t take the part unless he knew that he could do it justice. “So I said, ‘Listen to the score carefully and if you think you can handle it, fine by me – and I was right,” says Sondheim when I interview him later in London. “I knew he was not about to get up there croaking. So Johnny Depp cast Johnny Depp.”

The composer did, though, have casting approval and was sent audition tapes of all the hopefuls – Sacha Baron Cohen as flamboyant rival barber Signor Adolfo Pirelli, Alan Rickman as Judge Turpin, Timothy Spall as Beadle Bamford and even the director’s girlfriend, Helena Bonham Carter, who has been a huge Sondheim fan since she was a teenager.

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