Simon Godwin's production turns Malvolio into Malvolia, with Tamsin Greig playing the uptight steward. Malvolio has long been seen as the plum part in this comedy of mistaken identities, and Greig rises to the occasion fruitily. She has a gift for putting an audience in stitches with the tiniest face twitches; her expressions are their own Shakespearean asides. Malvolia's puritanical pomposity is fastidiously caught, but so is her gawky flush of love, and Greig nicely involves the audience (who adore her). It's a show-stealing performance – although only a step away from show-boating.

Given such high-profile cross-casting in a play where girls dress as boys to woo girls for their boys, you might expect this production to be strenuously making points about sexual confusion. In fact, Malvolia being in love with her mistress Olivia feels barely worth remarking on, and throughout sexual and gender fluidity is presented with a ‘yeah obvs' matter-of-fact-ness.

Antonio is demonstrably in love with Sebastian from their first scene. Sir Andrew might prefer Toby to his niece. At the end, Viola doesn't need to come back on in women's clothing: we get that Orsino rather likes her as a boy (so much so he accidentally snogs her brother, one last fillip of confusion). Even Olivia, often played as ecstatic to have bagged the male twin, is here sullen; she'd clearly rather have the boyish woman she fell in love with than a man she's only just met. It's a logical reading, supported textually by the fact Olivia never actually addresses Sebastian in this scene – even if it does quietly disrupt the double-wedding happy ending.

But then, the neat conclusion of Twelfth Night is never as cheery as the mayhem that's gone before, and Greig provides a particularly vulnerable, done-in Malvolia. Her promise to be revenged "on the whole pack of you" is more mournful than malicious, although it effectively extends to the audience. We were complicit, after all, laughing at her attempts to woo her lady in yellow stockings (and a retro yellow bathing suit).

Well, it was funny. And in general, Godwin's production is a hoot; while there's still plenty of impenetrable Elizabethan wordplay, he's assembled a cast of fine comedy actors that winkle out as much as they can, and fill the stage with daft dancing and fighting, strutting and wooing. As a sozzled pair of lushes, Tim McMullan's Sir Toby Belch is a dissolute dandy with a dry drawl, while Daniel Rigby minces about in hot pink as the dim Sir Andrew Aguecheek. Doon Mackichan – more cross-casting – gives the unfunny fool a sparkle that goes far beyond her glittery boots (she looks like she just crawled in from Secret Garden Party), and their EDM-pumping parties are a comic highlight.

Phoebe Fox as Olivia switches between pert propriety and head-losing lust – although her chemistry with Tamara Lawrance's Viola doesn't convince. In early scenes Lawrance's delivery feels strained, as if struggling to fill this vast venue, but she manages the character's bewilderment well, and private moments of cradling her crush on Orsino (Oliver Chris in smoothly entitled bachelor mode) are endearing.

The setting seems to slip and slide between the present and the 1930s, with smooth live jazz and Art Deco stylings. Soutra Gilmour's clever design folds out in a series of triangles: now an ocean liner, now an elegant conservatory, now a fetish club…

Still, this isn't the raunchiest or most radical Twelfth Night; there's no big revelation. Yet it is stylishly, confidently done, really unleashing the humour. And Greig's Malvolia proves gender is no barrier – either to playing great parts, or to being an utter fool for love.

Twelfth Night runs at the National Theatre until 13 May. An NT Live broadcast takes place on 6 April.