After a few days of will-they, won't-they online speculation caused by the deletion of their entire online presence, Radiohead have indeed returned with their ninth album, and first since 2011's The King of Limbs, A Moon Shaped Pool. As trad rock fans who gravitated to them early in their career have been moaning ever since Kid A in 2000, it's not exactly filled with stadium anthems. Yet as a cohesive piece of work which builds a unique mood while offering an oblique reflection upon modern communication and the effect it's having upon our society, it's an absolutely essential piece of work filled with resonantly beautiful songs. Nigel Godrich produces once again, with the London Contemporary Orchestra providing the album's copious string parts.

Burn the Witch
As far as comeback singles go, this one couldn't have told us any more definitively that Radiohead are back. That intro, in particular; a gorgeous, escalating, full-of-potential string orchestral riff, then a meatily growling synthesiser note and finally Thom Yorke's voice, pained but imploring as ever and beset by a certain undercurrent of difficult to articulate anger. 'Stay in the shadows / cheer at the gallows / this is a round-up,' he whines.

Those who caught the song at its debut, complete with the painstakingly recreated and uncharacteristically jovial Trumpton homage of a video, should be well aware of early interpretations of the song as a metaphor for fear and mistrust caused by the refugee crisis (Yorke talks of 'red crosses on wooden doors' and 'loose talk around tables', invoking plague victims and McCarthyism among his potent metaphors), but this song speaks of something deeper. Maintaining a dramatic urgency which hits a kind of frantic crescendo and threatens to derail its finely-poised purpose by the end, it articulates the heavily amplified mob mentality of discussion in the 21st century, the weird, dystopian self-censorship imposed by giving everyone a voice. 'Shoot the messages,' indeed. As album openers go, it's nearly a match for Hail to the Thief's 'There There'.

Daydreaming
The theme continues, but certainly not the sound, with this gentle, regretful lullaby built around a simple repeated piano arrangement, an eerily reversed vocal sigh in the background and Yorke's voice, as ever, tipping over into the abyss at the edge of the world. 'Dreamers / they never learn / beyond the point / of no return,' is his opening gambit here, and it's devastating; don't just give up, but don't try unless you make things worse. Yet the song doesn't sound defeated, more weary with the realisation that altruism alone doesn't necessarily make you right.

Decks Dark
In which that feeling of doubt articulated on 'Daydreaming' is exacerbated with a degree of unnerving beauty. Once again, Yorke's singular vocal takes centre stage over an absent-mindedly strummed electric guitar and a tender, wordless chorus, as he contemplates the darkness caused by a metaphorical spaceship filling the sky. There may be further nods to the nature and policing of online communication in 'it was just a laugh / just a laugh / it's whatever you say it is / split infinitive', but seeking stories behind this music only serves to stifle it. You're reminded of the films of Paul Thomas Anderson (not an unlikely touchstone, given he regularly collaborates with Radiohead guitarist Jonny Greenwood and directed the video for 'Daydreaming') – these songs aren't stories, they're morally ambiguous planets of their own, paintings of the world as it presents itself and not how their creator wants it to be.

Desert Island Disk
Unlike David Cameron's clearly committee-chosen inclusion of 'Fake Plastic Trees' on Desert Island Discs, it's possible to see someone honestly choosing this song as a life marker in years to come. As subdued as the previous two, it makes a shimmering volte face into the realm of almost entirely acoustic folk, with a sense of Nick Drake or Tim Buckley emerging as Yorke meditates upon being 'totally alive / in my spirit light.' The guitar takes a turn for the more jazz-infused midway through, but a sense of warmth and possibility never leaves the song. If 'Daydreaming' and 'Decks Dark' are the sound of a man solemnly switching off his tablet in resigned disbelief at the world, this is him escaping to the country on a hot summer's day with some non-prescription medication to escape the modern world.

Ful Stop
Back in the real world and standing on shifting ground once more, this is an understated goliath of a song. It builds ominously on a swirling, benighted synth groove and a lacerating sustained keyboard note, escalating the mood to a crunching, fragmented beat which is drone-like in its qualities. Yorke, breaking into an uncharacteristic holler, ponders how 'you really messed up everything' and 'the truth will mess you up'. It sounds more personal than political, a fragmented letter to a wronged lover, but again, context only ties these songs down. So far, it's the one which really makes you want to wring the listening out of it.

Glass Eyes
Tender, kitchen sink anxiety builds beneath soft, cinematic sweeps of string and piano, again fusing the two key themes (so far) of alienation from society and perhaps from an individual. The narrator alights from a train into an unfamiliar town, and picks his way through faces of 'concrete grey' and towards the mountain. It defies much more description than that; let's just say it's as hauntingly, humanly beautiful as anything this band have done.

Identikit
Yorke's echo-heavy, rhythmic vocal and the warmth of the bassline skipping around reverb-laden guitar swooshes reminds of another survivor of the more cerebral end of 90s rock, Fife's own Steve Mason. There is, unfeasibly, a certain low-key soulfulness here, infused with a very thin drop of reggae, as Yorke hollers that 'broken hearts make it rain' and contemplates 'pieces of a wreck of mankind that you can't create.'

The Numbers
Woah, positivity. And a lot of it. Again, amid a heavy dose of reverberating studio production, the folk edge is accentuated, but this time alongside an electric guitar which growls in the background and strings which make the hair prickle with their sense of purpose and determination. It's only a more commanding vocal from Yorke away from being a protest anthem for our times, bringing to mind the arrangements of John Martyn or Gil Scott-Heron as Yorke asserts 'the people have this power / the numbers don't decide … we'll take back what is ours.'

Present Tense
There are no fillers on this exceptionally well-conceived album, but this feels closest to it, a typical ambient Radiohead number built upon shuffling drums, swirling guitars and that cycling vocal trick Yorke used beautifully on 'How to Disappear Completely'. The choir is astonishing, however, as is the poetry in his words; in particular, 'distance is like a weapon of self-defence / against the present tense' chopped up and repeated as an opening verse of singular rhythmic potency.

Tinker Tailor Soldier Sailor Rich Man Poor Man Beggar Man Thief
The keyboard chime which sets the scene sounds like it's been stolen from an early 90s rave anthem, but slowed down to almost painfully somnambulant levels. Like 'Burn the Witch' and 'Ful Stop', with a lyrical thrust which is heavier on hard-to-read metaphor, this is one of the songs on the album which most precisely fuses musical ambition with thickly-layered atmosphere and that special kind of alchemy that a band like Radiohead have made their reputation with.

True Love Waits
And we've reached the end with no crescendo, no fanfare… just a truly affecting piano ballad. It's true that, as we've said, this band are best appreciated when the listener divines their own meaning, but it's impossible to hear these lyrics and not hope that everything's okay at home for Yorke; or it would be, if the song wasn't more than 20 years old. Many of the tracks here have enjoyed gestations going back a number of years, as evidenced by their appearance in past live setlists, but 'True Love Waits' has achieved a kind of legendary status as perhaps Radiohead's greatest lost song. Until now – it was first played at a gig in 1995, first released as a live version in 2001, and now it has its own definitive studio recording. As a plea to a lover to continue a relationship, it's supremely powerful; 'I'll drown my beliefs / to have your babies,' Yorke serenades us. 'I'm not living, I'm just killing time / your tiny hands, your crazy kitten smile… just don't leave.' After that, tears are permitted.

A Moon Shaped Pool by Radiohead is out now on XL Recordings.