The Stripper (St James Theatre)

Richard O’Brien and Richard Hartley’s murder mystery musical returns

Back in 1985, the team behind The Rocky Horror Picture Show turned their attention to pulp fiction with The Stripper, a musical based on a noir-ish 1961 whodunit by bestselling American author Carter Brown. Soaked in sex, Scotch and smokin' jazz, this revival in St James Theatre's small studio space offers much pastiche fun – but there aren't any inverted commas big enough to make the outdated sexual politics feel ok.

Al Wheeler (Sebastien Torkia) is a hard-boiled Californian detective complete with trench coat, trilby, and a follow-spot that allowed him to narrate his investigation in asides to the audience. A young woman named Patty fell to her death from the 15th-floor of a hotel – but, as an early song asks, did she jump, was she pushed or did she fall? In finding out, Wheeler gets caught up with various eccentric oddballs at a strip joint, a lonely hearts club, and, er, a florist.

The clockwork plot is sent up at every opportunity, in a show stuffed with cartoon grotesques and over-inflated sexpots. Richard O'Brien's bad-taste lyrics sail by on Richard Hartley's hot jazz score. It's larky and schlocky, but Benji Sperring's production has a tight control of the material and the heroically multi-tasking cast of only five have a real command of their many wildly OTT characters.

They all find moments of audacious or hoot-worthy humour, with Marc Pickering's flamboyant, shifty florist a highlight, and the chameleonic Hannah Grover particularly enjoyable as both an absurd Spanish strumpet and an icy, creepy matchmaker in owlish glasses.

But the soulful stripper supposedly giving heart to the tale is a cliché so painfully dated that it threatens to derail the show. There's surely no way to make the number where she sings about how stripping gives her a "sense of power" (yeah, really) work today without some kind of slant or comment. Weirdly, this is one of the few moments not heavily sent-up, and it comes right after the actress Gloria Onitiri really does do a full strip tease. Fierce and fabulous as she is, this is played uncomfortably – or at least unquestioningly – straight. There's no glimmer of the giddy, gender-bending subversion of Rocky Horror. And the male gaze isn't just privileged, it's goddamn spotlit and given a voiceover.

Sure, there are other moments when the heightened, almost aggressive sexuality of the women who trot around Wheeler is made comic – Grover is good at finding the daftness, and when it's very crude it can be very funny. A song about Wheeler's unrestrained libido is itself so unrestrained, it flies with terrible rhymes like "Your body's fantastic/I'll snap your elastic". But, especially in the first half, the relentless objectification becomes waring.

Tim Shortall's design makes the most of the postage stamp-sized stage, really too small for the show, but the cast work hard to stop you noticing. There's flashing neon and glitter a-plenty, and the (inter)action spills into the front rows, cabaret-club style. A band take one end of the stage – inexplicably not credited in the programme, they offer a swinging accompaniment, with rock'n'roll, doo wop and show tunes fleshing out the jazz score.

The Stripper does what it does with parodic aplomb. Still, I'm not convinced it needed reviving in 2016. After a few cocktails, it'll no doubt earn itself a cult following among ardent Rocky Horror fans, but – like the seedy floor show it sends up – examined in the cold light of day it loses lustre.

The Stripper runs at the St James Theatre until 13 August.