There are two kinds of band re-formation. The first is so compellingly straightforward that the “classic” bands that haven’t done it now seem weirdly anomalous. You bury your differences, a process eased by the passing of time, the sagacity that comes with age and, frequently, the promise of a whopping cheque: if the past 10 years or so have told us anything about musicians, it’s that few things are as effective at resolving those bitter, decade-long feuds over guitar overdubs or backstage catering arrangements or the drummer’s taste in wives as the prospect of paying off one’s mortgage. Then you rehearse, book shows, and knock out the hits, knowing the crowd will be so overwhelmed by nostalgia they won’t complain even if your singer sounds like a man who’s clambered on stage at a karaoke night after six pints, wrested control of the microphone and started bellowing down it, the Stone Roses having apparently reunited specifically to prove this.

The second involves actually recording new material, and seems infinitely tricky, fraught with the issues: not clumsily besmirching your own legacy, making music that identifiably fits with your back catalogue without merely appearing to pastiche past glories. Indeed, it’s proved tricky enough to bring reunions to an end: Kim Deal left the Pixies; the Stone Roses and Pulp clearly decided it wasn’t worth the aggro, while Jerry Dammers recently noted that his desire to record new songs was among the reasons he swiftly exited the reconstituted Specials.

So you can’t really blame the reformed Blur for stepping rather gingerly around the issue of adding a new album to an oeuvre, which, if anything, feels more highly respected now than it did at the height of their success: once drowned out by the distracting sideshow of their “war” with Oasis, the subtlety and breadth of the music they made in the 90s is easier to appreciate. Since re-forming to rapturous response six years ago, Blur were reported to have made three attempts to record a new album, but released only three songs: Fool’s Day in 2010 and Under the Westway and The Puritan two years later. Until recently, Damon Albarn insisted that the most recent sessions, in Hong Kong in 2013, had been a failure. No wonder a lot of journalists present at the press conference to announce The Magic Whip – a new Blur album, carved out of the Hong Kong recordings late last year by guitarist Graham Coxon and producer Stephen Street, with Albarn subsequently adding lyrics and vocals – thought the singer looked a bit surprised to be there.

It’s certainly an odd way to make a comeback album. But perhaps its peculiar genesis – born out of jam sessions, moulded into shape by Blur’s other protagonist – is what prevents The Magic Whip from suffering the fate of 2002’s largely Coxonless Think Tank: an album that had plenty of great songs, but felt like a Albarn solo project struggling to pass itself off as a band effort. The Magic Whip doesn’t contain anything obviously resembling a big hit, a fact Blur seem to have addressed by heralding the album’s release with the most tune-free thing on it – the feedback-and-amplifier-hum-drenched Go Out – there are plenty of moments that sound comfortingly familiar. Go Out could have hailed from the sessions for 13, the zippy I Broadcast would comfortably slot into the tracklisting of Parklife, opener Lonesome Street does a lot of things Blur were famous for when Adidas Gazelles were compulsory footwear for young men with guitars. It variously drop its aitches, mentions commuters on “the 5:14 to East Grinstead”, pays homage to Syd Barrett – there’s a hint of Pink Floyd’s legendary unreleased 1967 single Vegetable Man about Coxon’s vocal interjections – and has a go at consumer culture “mass produced in somewhere hot”: if it doesn’t feature someone shouting “Oi!”, it does feature some perky whistling.

This is all good fun, but The Magic Whip really comes into its own, in every sense, when it sounds least like music Blur fans will already own, when it most fits Coxon’s description of its contents as “sci-fi folk”. There’s a popular belief that one of the reasons Blur split up was that the band was incapable of containing Albarn’s increasingly eclectic musical interests, but The Magic Whip frequently suggests otherwise, finding a fascinating common ground between his and Coxon’s apparently divergent solo careers. Pyongyang finds the restlessly peripatetic singer boggling with horrified fascination at the North Korean capital over a weird, unsettling musical backdrop of tinny drum machine beats, vocal samples and organ; Coxon’s guitar sounds ghostly, it shivers and trembles, perfectly matching the lyric’s alienation. Mirrorball matches eastern-sounding strings to twanging, reverb-heavy guitar, while Thought I Was a Spaceman, the kind of delicate Bert Jansch-inspired fingerpicked guitar found on Coxon’s 2009 album The Spinning Top, rubs up against electronic noises and rhythms created on Albarn’s iPad, as they were on Gorillaz’s last album, The Fall.

Moreover, for music apparently jammed together during a few days’ downtime on tour, The Magic Whip is abundant in beautiful songs. Hazily lovely melodies wind through My Terracotta Heart, Ghost Ship and the peculiar stew of military drums, high-drama strings and vocodered vocals that makes up There Are Too Many of Us. New World Towers – more alienated boggling at foreign landscapes – is the kind of gorgeous, careworn ballad that’s quietly become Albarn’s signature style over the course of his solo work, a relative of everything from Hostiles on last year’s Everyday Robots to The Living Sea from his Monkey opera.

At least in part, The Magic Whip seems to owe its existence to Albarn’s dissatisfaction with Think Tank as the band’s final album-length statement: that sounded like the work of people at the end of their tether: “I’m here because I’ve got no fucking choice … Can we stop now, please?” Albarn sang on the track Me, White Noise. If The Magic Whip does turn out to be Blur’s final album, it’s certainly a nicer way of ending things, with its touching images of older, wiser men happily reconciled with each other and their past. There’s a lovely moment on Thought I Was a Spaceman, when Albarn dolefully remembers himself escaping to Africa, “digging out my heart, in some distant sand dune”. There’s a pause, then he mumbles “in Hyde Park” – the scene of one of Blur’s triumphant 2009 reunion shows – and the track suddenly lifts off, in a burst of euphoric synthesizer and woozy, My Bloody Valentineish guitar. And yet, for all its lyrical tying-up of loose ends, it’s hard not to hope The Magic Whip isn’t Blur’s last word. Musically, they don’t sound like a band taking a final curtain call. They sound like a band filled with ideas and potential new directions, who have plenty left to do together, if they choose.