Legion and lamentable are the reviews of female artists written by male music critics in which their hands appear to have drifted away from the keyboard for an energetic thigh rub, as their words collapse into panting cliché: "doe-eyed" singer songwriters, gothic electronic "vixens", delicate flowers whose achingly beautiful art the writer just cannot wait to protect in his sweatily cupped hands. FKA twigs is, therefore, potentially problematic, a honey trap for the ma... oh dear, there I go!

For it's nigh-on impossible to write about FKA twigs' debut album, and her live shows, without mentioning sexuality and physicality - the oddness of the intimacy she's able to create, the power dynamics within how she presents both her songs and herself in her role as performer. At Heaven a few weeks ago she dominated the stage in a catsuit as three nondescript men produced a tense, industrial racket on electronic drums and bass to the side and behind her. As a trained dancer, twigs' movements were always going to be commanding, but this was none of the arm-shoulder-headroll-shoulder-arm ripple, rictus grin plasticity you find on so many music videos and stages, the strangely automaton movement of everyday pop raunch. Instead, it felt as if twigs had taken the years of being an anonymous subject, as a ballet and contemporary dancer, backing dancer and burlesque artist, and twisted it into something powerful, feminine, dom, and - given that she told the audience that some of the moves were inspired by voguing - queer. Just look at that video for 'Hide': that anthurium flower's phallic stalk suggests not a hermaphrodite, but a strap-on. Oh bend over, boyfriend.

As her recent, excellent interview with The Fader suggests, so many of the inspirations for Tahliah Barnett's aesthetic world come from blurred sexualities, from the poster for Lolita and self-portraits of Frida Kahlo in her bedroom, to a video inspired by The Dreamers, the 2003 film involving a partly incestuous love triangle. I can also think of few pop stars who've been represented as a beyond-human, body-melted-into-mainframe cyborg on one hand and on the other used the ancient art of shibaru rope bondage in a shoot. Throughout twigs' work to date, this visual and thematic aspect has been interwoven with her approach to sound. Tellingly, in interviews she's described how the concepts for most of her videos are developed in tandem with writing the songs themselves, a further blurring process whose results draw attention to - and, if anything, heighten - the deeply peculiar nature of twigs' music.

Much has already been said about LP1 being a headphones album, a new future for pop, a fresh direction for R&B. I disagree with the first: I want to hear this on gigantic speakers, bass turned up, so it comes across as it did at that night at Heaven, something closer to the marshy rhythms of These New Puritans than anything on Radio One's daytime playlist. And as twigs herself has said, she doesn't feel that this is R&B. As she pointed out in a recent Guardian interview: "It's just because I'm mixed race. When I first released music and no one knew what I looked like, I would read comments like: 'I've never heard anything like this before, it's not in a genre.' And then my picture came out six months later, now she's an R&B singer." Instead, she exists in a space entirely of her own, which makes it important to consider this record as an entirely distinct entity from her past EPs and their videos. This is her direct appeal - this deeply unusual, insular, strange music, constantly and alternately giving and holding back, exposing vulnerability and commanding.

It's the dialogue between twigs' voice and the music that conveys this; a strange dance where both are not always exactly on the same page. As she has said, "I love annoying sounds, beats, clicks... It's got loud noises in there, the structures aren't typical, it's relentless". She's right: rhythms and synths pop, rattle, bounce, skip, ping, gurgle in strange time signatures. 'Pendulum' is built from creaks like ship rigging. before a terrific chorus rises like an unexpected double sun over the horizon. On 'Closer' there's a ponderous bass bong, fireflies of sound, and a vocal line that has the simple, communal feel of a hymn. 'Preface' is just one of many songs where, like much music at the moment, twigs recalls Enya in her something-from-beyond-the-night chants. Thrillingly, LP1 gives any record you might find us covering elsewhere on The Quietus a run for its money in terms of oddness.

"I/you don't need a man" is hardly a new trope in female-fronted pop, but how frequently is it put that masturbation is superior, as she does in album closer 'Kicks'? "I don't need you / I love my touch / Know just what to do". 'Hours' hides all sorts alongside the sweet refrain "I could kiss you for hours". It becomes "Am I suited to all of your needs, master?" before twigs herself does the biting. 'Lights On', one of her strongest tracks to date, slides into its sumptuous chorus from very little at all, just gun metal crunches, sparse patters, a vaguely glockenspiel-sounding plink. The clipped effect on her voice, the slight artifice of her breathy hiss - "when I trust you we can do it with the lights on" - feels like anti-autotune, a determined shift away from the abrasive, distinctly plastic textures that have dominated the foreground of so much recent pop music, towards something where the singer retains control over the manipulation of voice and self, and disregards the demands of radio, labels, producers and the loudness war.

The range of her voice, its crispness and clarity means you can tell where this is from, too - Bethnal Green from Gloucestershire via Croydon. On 'Girls From The Video' where "chance" has a pronounced, vaguely hoity-toity drawl on the middle vowel. "Why you gotta go make me cry" leaves out the 't' of the central consonants. That track, which references twigs' own past as a dancer, is however also the album's one weak spot. It feels strangely prosaic, over-emphasising a point already made more subtly elsewhere on an album that explores so many internal, sensual worlds. That aside, this is an album within which to disappear, where to disappear means to surrender. Surrender, of course, doesn't have to be a negative.

You only have to hear the word "thighs" being censored out in 'Two Weeks' on radio stations (alongside, of course, many, many more examples) to see the fear with which empowered female sexuality is still treated by the music industry. On a very simple level, this is why twigs is an incredibly important part of the musical firmament now, and will be in years to come. But equally, in some strange way I see FKA twigs touching on some of the threads that Cosey Fanni Tutti laid down in the late 1970s and early 1980s. Listen to the narrative title track of Cosey's wonderful 1983 album Time To Tell, about her time as a stripper and adult model, and compare it to 'Closer', or some of twigs' words in interviews. We exist in a pornographic age, so what Cosey was doing in back rooms and in the pages of porn mags then is now all around us, part of the very fabric of the distribution and marketing of what we know as pop music.

FKA twigs has managed to explore this existential condition of the performer, succeeding in accepting, diverting, turning and owning the male gaze. At the same time, she explores the strangeness of the dom, sub and switch that exists in all relationships, as they sit on a sliding scale between old-time monogamy, the letters pages of Cosmo and the hardest BDSM. That she's done this on such an uncompromising and weird album - and one which is now flying so far into the mainstream - is surely one of the most exciting things to happen in pop music for quite some time.