Meow Meow's Little Match Girl, Queen Elizabeth Hall

“How I do love a steely sting in my fairytale ending,” croons Meow Meow, eyes glinting even more brilliantly than her eyeshadow. When she says “sting” a whole army of scorpions couldn’t equal her venom.

As the title of this veteran “kamikaze cabaret” artist’s show makes clear, Meow Meow’s The Little Match Girl is an entirely idiosyncratic take on Hans Christian Andersen’s classic story, a piece of iconoclastic, bra-baring (if not actually burning) revisionist theatre – a “73-minute showbiz extravaganza on child poverty and social disenfranchisement.” Phew.

It sounds like a lot to pack into just over an hour, but the show is deceptively slight, alluding to much but never lingering on any of it. Weimar cabaret and Kurt Weill, Noël Coward and Cole Porter, Serge Gainsbourg and even Purcell find their way onto the stage, but all the razzle-dazzle sets and big musical numbers are just rugs for this canniest of artists to pull out from under her audience. Even the most heartfelt of ballads keeps a cynical shrug in store for its close.

A show that brings you to the brink time and again only to step coyly back

Essentially a one-woman show (with a little help from her friends), Meow Meow’s anarchic creation relies heavily on structured ad-lib and audience involvement. A lengthy bit of introductory business with the lights finds the Queen Elizabeth Hall in darkness, hapless audience members press-ganged into giving up their phones and shining torches on the star to enable the show to go on. Some poor bloke has to pedal a bicycle and a small group of people gamely wave their arms around – a human wind-farm. It’s all immaculately handled and stops just short of outright cruelty.

In between these frantic episodes Meow Meow shows off her serious vocal skills in a carefully curated mixture of covers and original songs, amplifying her theme in a collage of images and performance fragments. You’ve got to love the breadth of this show, its smart and endlessly referential self-fashioning and positioning. “I’m the Mother Courage of performance art” she quips, and half the audience laugh; she bends over to reveal a generous helping of underwear-clad bottom and the other half roar. There’s something for everyone in this decidedly adult pick and mix.

Anna Cordingley’s sets surround Meow Meow with chandeliers and glittering fringed curtains, but Paul Jackson’s lighting (or lack of it) brings the performance tightly in to just a face and a song. Juxtaposition of contrasts – rich/poor, warm/cold, tragedy/delight – are at the core of Andersen’s tale and at the centre of a show that brings you to the brink time and again only to step coyly back, or else career so swiftly over the edge that you find yourself in another world even before you realise you’ve left the old one behind. It’s dizzying and often incoherent, but rarely less than engrossing.

Yet it’s hard to lose your heart to something so knowing, to surrender when you know your tenderness will be briskly rebuffed. Meow Meow giveth and she taketh away this Christmas-time, and her fairytales are always going to be more Angela Carter than Andersen. There will be those that adore this emotional rough and tumble, the theatrical kiss-chase that teases just out of reach, but while it’s electric in the moment, the show’s performance-driven energy is short-lived, dissolving on the tongue to leave behind, what? A chalky aftertaste of satire; the sweat-smell of agitprop.

In a town full of “he’s behind you” panto this is Brecht and Borges, Tarkovsky and Tolstoy. But as Meow Meow leaves the stage a giant neon sign is illuminated: “Whatever”, it reads. Catherine Tate might not have been “bovvered” but you get the feeling that this show, this artist, really is. So occasionally, just occasionally, it would be good to see that, even if like the flash of secret flesh, the momentary reveal of burlesque, it can’t last.

• Meow Meow's Little Match Girl is at the Southbank Centre until 30 December 2012