If you've ever wondered what would happen if you transplanted the method of Captain Beefheart and the madness of Daniel Johnston into the gigantic papier-mache head of Frank Sidebottom (and frankly, who hasn't?) then this surreally comic – and yet poignant – oddity has the answers. Investing the frame of Chris Sievey's madcap creation with the tortured soul of avant garde rock, Frank manages to get beneath the mask and the skin of its eponymous antihero in a manner that bridges the gap between absurdist laughter and all-too-tender tears. The result is something weird, wonderful, and utterly unique – a cracked classic which takes its place alongside the Barbie-doll animation Superstar and the conjoined twins mockumentary Brothers of the Head in the pantheon of genuinely unexpected pop movies.

The roots of Frank lie in a newspaper article by Jon Ronson detailing his time as keyboard player in Frank Sidebottom's Oh Blimey Big Band. As with The Men Who Stare at Goats, Ronson's real-life reportage provided the springboard for a screenplay (co-written with Peter Straughan) which spins fantastical tall tales from stranger-than-fiction fact. To be clear: this is not the Frank Sidebottom story, in the same way that Todd Haynes's I'm Not There was not a Bob Dylan biopic. Rather, it inhabits an alternative universe in which mimicry and tribute (the film is dedicated to Sievey) form their own kind of strangely sincere (un)truth; in which characters try on one another's clothes, haircuts, and heads while striving to be somebody else; and in which it's not entirely unusual for someone to be sexually attracted to mannequins.

The film's closest link to "reality" is the faux naif dorkishness of Domhnall Gleeson's wannabe pop star Jon Burroughs, a Ronsonesque narrator who winds up playing keyboards for the unpronounceable Soronprfbs after the previous incumbent attempts to drown himself in the sea. Summoned to Ireland, Jon finds himself a willing prisoner in the rehearsal and recording of the band's new album, a year-long process which nods to legendary tales of Trout Mask Replica. Entranced by the fake head that group leader Frank wears 24/7 ("Would it help if I said my facial expressions out loud?"), Jon becomes seduced both by the guru-like enigma of his mentor, and by the waving hands of theremin player Clara (Maggie Gyllenhaal), who seems able to conjure beautiful sounds and savage weapons out of thin air with equal ease.

Getting the performers to play the music for real pays dividends, with composer Stephen Rennicks leading the cast through songs that range from the oblique (such as Lone Standing Tuft, an ode to a stray carpet strand) to the hilarious (Frank's Most Likeable Song – "people will love it") to the heartbreaking (I Love You All, already a haunting indie classic). From the chaos of an opening gig in an English coastal town, to the anarchic creativity of rehearsals in Ireland and the last-minute "new direction" of a terrifying appearance at SXSW in Texas, the assorted Soronprfbs look and sound like a "real" band trapped in a world of surreal strife. While individual characters may push the envelope of eccentricity, the horrors of a band on tour are as recognisable as the home truths of Spinal Tap (you can almost hear David St Hubbins exclaiming: "Too much fuckin' perspective").

Bearing the heaviest burden is Michael Fassbender, released (rather than weighed down) by the cumbersome mask which proves his character's liberation. Infusing the pantomimey finger-fiddling of Frank Sidebottom with the pathos of Daniel Johnston's twitchy tactility, Fassbender conjures a fully rounded character (from America rather than Timperley) whose expeditions along the edge of sanity question oft-repeated cliches about the genius of madness (and vice versa). With his facial expressions obliterated by Frank's immovable look of ooh-er astonishment (and his voice muffled by eggshell casing), Fassbender is forced to speak with his body, his stance and mannerisms precisely tuned to the complex tragicomic twists of the story.

It takes a director of some talent to stop such an outlandish venture tipping over into mere quirky indulgence, and Lenny Abrahamson rises to the challenge magnificently. From the Beckett-inflected riffs of Adam & Paul to the domesticated sociopathy of What Richard Did, Abrahamson has proved himself an astute observer of "borderline" behaviour. It's hard to find a duff performance in any of his films, and false notes are few and far between. Here, he orchestrates the key change from goofiness to sadness with an elegance that means we never notice the sly introduction of a narrative minor-third. As Frank peels away the layers of its subject's onion-like facade to reveal a piper at the gates of dawn beneath, so our own grinning expressions start to falter. And whereas a lesser director would have been content to leave us amused and amazed, Abrahamson ensures that we are also moved.

While Frank may not be for everyone (just as Sidebottom, Beefheart and Johnston were never chart-toppers), for those who like their movies to dance to a different beat, it is something rather exceptional.

You know it is. It really is.