After Blur's 6 Music gig – the first of two shows they're playing for BBC radio tonight, to begin what they have suggested may be their last fortnight as a band – their 1995 single, The Universal, plays on the air. This feels awkward. Graham Coxon, head down like a teenager, is messing with his guitar pedals. Bassist Alex James and drummer Dave Rowntree look sheepish. But Damon Albarn is listening, and smiling at the audience, then mimes along to his band's usual set-closer, conducting the crowd. "It really, really could happen," they sing, as he waves his hands.

But could it? In the run-up to next week's Olympic closing concert in Hyde Park, there is a fear that Blur might have "made it to the end". This has been their week of tiny-venue rehearsals all around the UK to hone their intimate connection with the crowd before their swansong goes super-sized. Their box set, 21 ,is in the shops, bringing an air of finality with it – and it doesn't feel right. But why? Perhaps it's because they're the closest we've got to an intensely British heritage act, hence their role as Olympic Games curtain-closers. Everyone else has retired (David Bowie), hasn't reunited yet (the Kinks, Oasis, the Jam) or has long seemed a global entity (the Rolling Stones). Or maybe it's just nostalgia for the wayward mid-90s, a time when songs about the shipping forecast (This Is a Low) and heroin use (Beetlebum) became festival anthems.

Tonight's two gigs see Blur revisit their past properly. The 6 Music show begins with 1994 album track Jubilee, a tale of a lazy 17-year-old (and a nod to recent events in name only). Albarn's on showman-mode straightaway, instantly younger than his 44 years. The band also play urgently, backed by two meaty trombones. Proof that they've been working hard comes in the shape of two barely played B-sides: their second-ever performance of 1993's Young & Lovely and 1991's Mr Briggs. Before the former, Albarn explains it was written about their parents looking at their children, but now they occupy that role. "Somehow it's come to life," he smiles, and the song soars with extra sentiment.

Then we bounce back to Blur's present, and their two new tracks. The Puritan references the "flash of a blade" and the anaesthetising effects of "dry ice" and "autocue" on TV; it's far from their best song but it fizzes with menace. Its epic flipside, Under the Westway, should sound like a parodic goodbye. Blur's much-loved London is sinking, its skies are on fire, the Last Post sounding "like a love song". This song sounds like Procol Harum's A Whiter Shade of Pale, certainly, but its lyrics also carry a whiff of JG Ballard. Blur were always capable of such literary ambition.

Tonight's best moment, however, doesn't come from social commentary but memoir. For the first time, they play Caramel from 1999's 13, a stunning, sonically inventive sad song about Albarn's break-up with Elastica's Justine Frischmann. "I want to get over... I want to get better," Albarn sings, before all hell breaks loose under Maida Vale's wooden fittings. James faces Rowntree, both pummelling away. Coxon is blissfully lost in the noise. I'm reminded of REM's final tour, where that band revisited so many lost songs. There's a similar joy here, a spirit of exploratory fanfare, that goes nuclear in Popscene, the fan-favourite 1992 single.

At the end of the Radio 2 In Concert session, The Universal arrives properly. It remains a clever, layered set-closer. A warning about the next century, where "every night we're gone... to karaoke songs/ And how we like to sing along". You look around, and here we are, joining in on the verses. Albarn looks lost in the moment, fluffing a line at one point. But when the last lyrics arrive, they get these fans in the gut: "When the days they seem to fall from you, well, just let them go". How hard it is to let Blur do that on this brilliant form.