It's the final day of July and, even if the temperature's tapered off from replicating a citywide pressure cooker, it's still hot and close. Descending into a subterranean spot crouching between railway arches is a venture at best reserved for the foolish or the maximally refreshed. Nonetheless, Heaven feels right as a spot for watching FKA twigs close a short run of shows she's playing before the release of her debut album LP1; sealed off from summer hum, sultrily uncomfortable, dark, airless, it's the ideal quarter for twigs' weird, intimate music.

She walks out onto a backlit stage that leaves her in silhouette. The lighting adds a sort of authority to her poses, emphatically burnishing the Destiny's Child-like sashays, the voguing and impassioned shapes thrown, while simultaneously throwing the audience under a silencing veil of darkness. Her gold-plated jewellery looks of a piece with Nabil's video for LP1's lead single, 'Two Weeks', where twigs gets enthroned as empress of an ancient, distant court, heat-strewn (flies buzz in the steamy air in the background) and filled with dancers in constant, mesmeric motion, and she seems to occupy both roles tonight.

Musically, her calling card is her songs' shard-like quality, where different generic planes rub against each other. There's the trappy 808 hi-hat trills on 'Give Up', aqueous ambience on 'Water Me', industrial percussive clatter on 'Pendulum'. Its variety calls up, weirdly, Oneohtrix Point Never, especially moments on last year's R Plus Seven, where glints of vocals, tuned percussion and panpipe suddenly fell away to vertiginous gaps of silence.

But all that's essentially subjugated; LP1 is scattered with odd sonic insignia that make it sound pretty much unlike anything else out there (and have similarly transformative effects for others too, as on the Lucki Eck$ track she recently produced), played with visceral precision by three musicians on pads and bass tonight. After an opening pair of songs from her first EP, the mutant guitar of 'Lights On' slinks across the speakers, leading to a chorus that pans across a vista of handclaps hitting in and out of time, with the resonation of a double bass quivering somewhere beneath twigs' repetition-to-incantation, "When I trust you we can do it with the lights on", before the song closes with the blindsiding brazenness of a car alarm.

For all the disparity of their components, the songs overflow with hooks. The album in particular works like some elaborate hoodwinking, pulling you in with this curious mass of sounds and styles and lulling you into a lotus-eater daze before the veil lifts, and you're left with a head full of voracious earworms. Midway through the set, the synthetic ululations of 'Water Me' give way to winding-down metronomic clicks, which twigs physically syncs to, moving with clock-hand rigidity, before the chords, formed from echoes of her pitched-down voice, ricochet off the walls, leading into EP2's centrepiece of desire-burned melancholia.

At the gig's three-quarter mark, the hyper-treated vocals of 'Two Weeks' shake Heaven's walls. There's an audible wave of appreciation: this is the album track that's been in the public consciousness for longest, and maybe even the LP's most immediate moment. Still, though, you could listen to 'Two Weeks' ten times in a row and not feel even faintly tired of it. Broken down to its parts, it almost feels scant; the bass tremors and strobing synth patterns that cycle over are among the album's sparsest backings, but lined with twigs' words, an imploring to fuck that embodies itself in straining, physical metaphors, and vocal line, cut-up phrases building to airy, octave-vaulting layers, it becomes a song that's cast with an honesty and directness that's infrequently heard. Tonight, the groaning bass strikes blows to the chest and surges of white light at the chorus render the track, an outright euphoric, half-speed banger, in sharp relief.

Twigs' voice can't quite keep for the constant leaps of 'Papi Pacify' and 'How's That', but no one cares by now. The immersive, addictive world that FKA twigs creates is already well into its process of realisation and when she leaves after a little over an hour, the feeling, one and the same with that which you experience once the final reverberations of 'Kicks' ring out on the album, is the need to make immediate return.