It increasingly feels as if trying to nurture an atmosphere of mystery in pop isn't worth the bother. For years, it was taken as read that pop music was a kind of theatre of dreams in which people could reinvent themselves. Audiences didn't ask too many searching questions; a certain mystique was self-evidently part of the fun. In the age of social media, it doesn't work: however long you spend cultivating an intriguing persona, someone is guaranteed to swiftly pop up gleefully brandishing the mundane truth about you. When they do, it can damage your career. For reasons that aren't entirely clear, people have recently shown a marked tendency to get all how-dare-they? when an artist is revealed to be pretending to be something they're patently not, as if they think pop music should be covered under the Trade Descriptions Act.

It perhaps says something about the power of FKA Twigs' music that her standing seems to have been unaffected by the discovery of the prosaic figure behind the enigmatic press shots and adventurous videos, the shock revelation that her records aren't really made by a shape-shifting cyborg or a dead-eyed mannequin or indeed a kind of human anime character. In fact, they're the work of 26-year-old Tahliah Barnett, a dancer from Cheltenham whose previous brush with fame involved appearing in the videos for Do It Like a Dude and Price Tag by the legendarily enigmatic avant-R&B auteur Jessie J. Moreover, enthusiasm for her work seems to have been undimmed by the sub-genre of alt-R&B reaching a kind of saturation point. From the Weeknd's troubled and troubling reinvention of the priapic R&B loverman, via the divas hooking up with experimental dubstep producers, to the countless indie artists knowingly dabbling, there's been an awful lot of it made in recent years. Even its most dogged adherent must now have enough tracks on which an etiolated chillwavey synth washes over a Rodney Jerkins-influenced beat to last them a lifetime.

That FKA Twigs' releases to date have been met with excitement rather than ennui tells you a lot about how singular the music she makes is. It's not just Barnett's fondness for Björkesque visual presentation that recalls the late 90s: what LP1 really invokes is a radically updated take on Pre-Millennium Tension and Angels with Dirty Faces, the dark, brilliant, career-knackering albums Tricky made while literally maddened by a combination of drugs and candidiasis, an infection brought on by asthma medication. It opens with the kind of choral singing that normally heralds imminent death in a film about demonic possession, rather than an album full of R&B slowjams. Whether cooing or moaning or drenched in effects, Barnett's voice always sounds distinctively British: she shows off her vocal chops not by indulging in melismatic showboating, but by swooping into a high, choirgirl-like register. The arrangements short-circuit, lapsing into dischord or silence; disconnected sounds suddenly arrive out of nowhere and vanish just as quickly; the beats occasionally clatter out of time or appear so scattered and sporadically that it's hard to grasp exactly what's going on rhythmically.

As you might expect, this approach works best when there's a strong melody at the centre of it all. At its least appealing, as on Numbers, LP1 sounds like a load of quirky sonic ideas scampering about in desperate search of a song to cling to – there are moments when the sudden bursts of noise and dischord sound irritatingly intrusive. But when the tunes match the invention of the production, LP1 is genuinely brilliant. The chorus of Lights On gleams brightly through the disorientating clatter, Two Weeks sounds thrillingly like a hit record that's being allowed to unravel before your ears.

The one similarity between Barnett's work and that of the Weeknd is their shared interest in subverting R&B stereotypes: while the Weekend casts the amoral, moneyed "playa" in a disturbing new light, Barnett's songs offer a distinctive take on the traditional female roles of seductress and wronged woman. The sirens she portrays are frequently confused and vulnerable. Their sexual assertiveness is underscored by self-doubt, which seems a pretty realistic depiction of sexual assertiveness, regardless of gender: "When I trust you we can do it with the lights on." Other times, they seem faintly terrifying, lust bordering on obsessiveness. "I could kiss you for hours," she sings, her voice gradually slowed down until it sounds like a threat: you're not sure whether the recipient of her affections should willingly submit to her charms or get their number changed. "How does it feel to have me thinking about you?" offers Pendulum: it should be a straightforward come-on, but something about the musical backdrop – muted guitar, rattling electronic percussion, a creaking noise that sounds like wood about to splinter, her voice high and ghostly – makes her question seem unsettling. How does it feel to have you thinking about me? Um, can I get back to you on that?

Her wronged women, meanwhile, aren't resilient I-Will-Survive types: they sound utterly crushed. "You lie and you lie and you lie … I can't recognise me," complains Video Girl, before the song grinds slowly to a halt, as if collapsing entirely. The abandoned protagonist of the closing Kicks elects to take matters into her own hands, so to speak. "I don't need you, I love my touch, know just what to do, so I tell myself it's cool," sings Barnett, bringing 45 minutes of confusing, fumbled come-ons and romantic disappointment to an impressively bathetic conclusion by giving up and having a wank instead.

It almost goes without saying that not many albums of any genre end like that. But then not many albums sound like LP1, a singular piece of work in an overcrowded market. It has its flaws – as you might have intuited from the videos and press shots, they largely stem from trying a bit too hard – but you leave it convinced that FKA Twigs is an artist possessed of a genuinely strong and unique vision, one that doesn't need bolstering with an aura of mystique. Given the times we live in, that's probably just as well.