Of all the bands to have emerged from late 60s rock, King Crimson are one of the least storied. There are no anecdotes of bat-biting, telly-defenestration or naked bicycle races. This perhaps reflects the sober personality of their only constant member, guitarist Robert Fripp, who's overseen events from his stage-left stool since 1969, not so much the leader as the curator. Tonight he's in customary three-piece suit and horn-rims, rocking a benign-but-impassive-bank-manager vibe that's very Gary Oldman in Tinker Tailor Soldier Spy. The whole band is dressed similarly, and the absence of showmanship extends to the presentation: no talking to the audience (although they do smile now and again), no jumping about (to be fair, their average age is 63) and a tasteful but mostly static lighting design more reminiscent of an IKEA showroom than a rock concert.

Why haven't more people heard King Crimson? They've had many line-ups, and have hardly courted mass success. And in their first set in Glasgow, it becomes clear that they're not exactly in an ingratiating mood. We are greeted by Fripp's polite Dorset tones over the PA, respectfully asking us to refrain from taking pictures until the end, but apart from that, the only active crowd interaction is when woodwind player Mel Collins inserts 'I Belong to Glasgow' into a solo, yielding a polite ripple of applause – an unexpected if not unprecedented moment of pure cheese.

What other veteran band in 2018 would play a good hour of some of its least-liked music, and dramatically improve on the original versions? 1970's Lizard was an awkward and rather stilted album of un-catchy pop and gloomy Mellotron-driven prog, and yet the 2018 line-up plays the entire second side and gives it an emotional power that the original recordings just don't have. A song like 'Cirkus', in the mouth of singer/guitarist Jakko Jakszyk, is finally as affecting as it was clearly intended to be. Fripp's mournful, microtonal solo on 'Prince Rupert's Lament' is more like a pibroch than ever. It becomes clear why King Crimson have filled concert halls rather than stadiums: at their considerable best, their music can be a right downer. The despairing 'In the Court of the Crimson King' is not a political song, but in the age of the Court of the Orange Clown, it speaks to everyone. At the end of the first set, 'Islands' offers a little catharsis, with some lovely piano from drummer Jeremy Stacey.

The second set shakes off the melancholy and gets more physical. 'Neurotica', in the absence of its original singer Adrian Belew, becomes a dizzying jazz-rock workout. Some songs are broken down into their components and put together differently: an edgy, strangely familiar stop-time riff, oddly evocative of Pharrell Williams' theme song for Despicable Me, cues a brilliant improvised drum trio in which Stacey amusingly tries to throw the whole thing off by playing in a completely different tempo and metre – but then that riff turns out to belong to 1981's 'Indiscipline', transformed from its spoken-word original into a mixture of roiling, off-kilter rock and yearning jazz ballad. 'Larks' Tongues in Aspic Pt 2' showcases the majesty of bassist Tony Levin, whose gleaming bald pate was paid homage to by the gentle, rhythmic bobbing of all the bald heads in the stalls. (Crimson fans like me are always absurdly gratified to see anyone at a gig who's clearly under 30. Tonight, there are actual young couples in the audience. Well, some, anyway.)

Set-closer 'Starless' sees the evening's only lighting change, as the band is slowly bathed in hellish red light which … oh, it's crimson. Duh. The encore is the only KC song that tends to get played on the radio: the furious, 1969-vintage '21st Century Schizoid Man', timelier than ever, which gets the crowd singing along and still finds time for a blistering and witty drum solo from Gavin Harrison. (They've had three drummers for a while now. Why? Because one's not awesome enough.)

Even when revisiting their back catalogue, King Crimson defy the tendency of elderly rock bands to become their own tribute act. They re-attack old songs with an exhilarating sense of What would happen if...? So many bands get mummified in their own past, but this one roars with life.

Reviewed at Glasgow Royal Concert Hall, Mon 12 Nov. Edinburgh Playhouse Tue 13 Nov.