When The Rolling Stones played at Twickenham in the summer of 2006, I was lucky enough to bag a seat relatively close to the stage. Close enough, in fact, that I could watch Mick Jagger’s extraordinary contortions without having to rely entirely on the big screens.

Marvelling at the band’s enduring excellence, I was also transfixed by Jagger. It was hard to think of another artist who palpably concentrated on their performance quite this hard; not their playing or singing, but their physical performance. Every single move, whether choreographed or spontaneous, seemed to be the product of rapid, narcissistic, intense calculation. It was impressive, compelling and a bit weird, too.

I was reminded of this, anyway, watching Martin Scorsese‘s new film about the Stones, “Shine A Light”. “Shine A Light”, as you may know, is a concert movie more or less, filmed at New York’s Beacon Theater in the presence of Bill and Hillary Clinton, Hillary’s mum, their guests, a selection of enthusiastic beautiful people ringing the stage and, we can only assume, a bunch of authentically grizzled Stones fans further back in the venue.

For a couple of hours, Scorsese’s battalion of cameras tail the band around the stage, nuzzling up close to these remarkable men as they go about the business of a lifetime with commendable vigour. As a portrait of how a band can grow old and make us rethink how we are expected to grow old, it’s fascinating – not least because the reliably wily director intersperses the action with archive footage of the band being interviewed in their pomp.

But without taking too much away from the sterling playing and slouching of Keith Richards and Ronnie Wood, and the tidy pragmatism of Charlie Watts (as he did at the end of the two shows I saw on the last tour, Watts zips on a sensible fleece before he steps out to take a bow), it strikes me that this is a film about the strange miracle of Mick Jagger.

You could view much of “Shine A Light”, in fact, as an almost fetishistic celebration of Jagger’s body: the relentless close-ups on that lined face and neck; the logic-defying contrast with that dynamic teenage body. Scorsese is clearly almost as fascinated by the spectacle as Jagger is himself, lingering while the singer lifts up his shirt to reveal the flatness of his stomach; capturing the frightening concentration that I mentioned at the start.

There’s always a lot of talk, when the Stones are mentioned, about either their love of performing to a crowd, or their love of making unimaginable amounts of money. Watching Jagger, though, something else seems to be going on. He seems to be in another place entirely: you don’t get the impression he feeds off the crowd’s energy – his interaction with them is pretty tokenistic – but his focus is total. I’m not sure it’s as simple as proving something to himself, either. It’s a movie, and Jagger is kind of acting, so it seems reasonable to ask the question: what, exactly, is his motivation?

And the answer, of course, is that I don’t know. Perhaps the most profound thing that Scorsese uncovers in “Shine A Light” – other than that the Stones remain a phenomenal rock’n’roll band – is that what has kept them going so long isn’t just something as straightforward as greed or obsession, but also something baffling, intangible. There’s a lot of enjoyable roleplay at the start of the movie, especially, when the band slip into commercially exigent stereotypes – Keith the easy-going louche, playing pool with Ron; Jagger the uptight, meticulous control freak micro-managing every aspect of the production.

Scorsese milks this for some amusing drama involving himself, with the director going through a pantomime of trying to get the setlist out of Jagger before the gig (it’s a capricious selection, incidentally, perhaps designed not to reproduce the content of previous Stones live product). He films the show beautifully, from the hook-up between Buddy Guy and a plainly awed Stones on “Champagne And Reefer”, through to an exhilarating set-piece of “Sympathy For The Devil”. Live films often bore me, but “Shine A Light” is pretty gripping, even during its weaker moments (“Loving Cup” could manage without a surprisingly overwhelmed Jack White, for instance).

But I guess ultimately, Scorsese has found one of those romantic, mythological, quintessential Scorsese themes buried here: that however obvious the motivations of men might seem to be, their calling can sometimes be more powerful and mysterious than we can easily comprehend. And, of course, a calling that can be utterly and completely without end, too. Good film.