Let's do the concert right here

They turn to play, facing each other like vying guitar players in a rock anthem. They beckon to the assembled crowd to follow them inside an empty theatre and proceed to give a performance of the whole work.

There have been similar scenes since then: on a boat back from Ischia, in Mannheim tram station, in a supermarket in Frankfurt. Beethoven, Schubert, Mendelssohn and Bartók have been performed. The pocket orchestra always appears spontaneously, the cellists using extra-long spikes to steady their instruments while standing — or finding somewhere to perch.

There's a different line-up each time but with many of the same faces, strikingly young and smiling yet performing with the precision and panache of the great symphony orchestras of which most are also members. This is European classical music's latest phenomenon: Spira Mirabilis.

"We're not trying to be cool," says the orchestra's charmingly earnest co-founder, Timoti Fregni, who insists these guerrilla performances are no PR stunt, more a manifestation of their overflowing enthusiasm for the music. "Our mission is not just to entertain, or to raise money, or for a social aim. It's music for music's sake, and we're convinced it makes sense."

Despite — or perhaps because of — these protestations, Spira have wowed critics and audiences across Europe. This week the group heads to London having attracted the attention of the Southbank Centre's head of music Marshall Marcus, a founder member of the Orchestra of the Age of Enlightenment (OAE), which was also set up to challenge the classical status quo.

Marcus hails their "idealistic lunacy" in seeking to reinvent the concert — like the OAE, Spira Mirabilis like to talk their audience through the piece before they play it — and enthuses about how rare it is for young Europeans to play with such fervour. He sees them in the same vein as Venezuela's youth orchestras, and has programmed them for the second concert of the Shell Classical International series at the Queen Elizabeth Hall.

The choice of the Eroica Symphony is especially apposite for the passion it conveys in a conventional form. "I promise you three great revolutions on Friday: Beethoven, Spira Mirabilis and Guy Fawkes," Marcus beams.

Spira was set up three years ago by violinist Fregni and his friends, Lorenza Borrani, Giacomo Tesini and Miriam Caldarini, in Fregni's home town of Quercianella in Italy. They hoped to reinterpret music with more time and space for rehearsal than was possible in the established orchestras in which they play.

There was no name when they gave their first guerrilla performance — locals just called them the "Neverland Orchestra" — but they landed on Spira Mirabilis, after the Nautilus shell geometric form, whose shape stays the same however much you extend it. They felt it reflected the way the group would remain true to its musical principles however much it changed in size and scale.

Next followed an appeal to towns across Italy to host them — and an answer came from little Formigine near Modena, where the authorities offered basic lodgings for 25 and a hall to rehearse in, and which has become Spira's base.

Over time, the original group has collected more talented musicians from friends of friends in the wider European concert scene. They include the strikingly expressive young German cellist Luise Buchberger and the OAE flautist Giulia Barbini, though they are all at pains to keep their day jobs separate from what they do with Spira.

Two Brits have performed with them: horn players Tom Kane from Northern Ireland and Katy Woolley from Exeter. The youngest current member, Ursina Braun, is an impish 18-year-old Swiss cellist. The oldest, bass player Juan Sebastian Ruiz, is 38 and trained as a mechanical engineer in his native Panama, seizing the chance to embark on a professional music career through a scholarship to a German conservatoire.

Handsome young Breton Nicolas Fleury, who studied at the Royal College of Music, describes the pleasure of playing music with friends who go off for games of football between rehearsals, as opposed to "being paid to play your notes" in a conventional orchestra.

The rehearsal process is democratic: there is no conductor so anyone can raise a query as the orchestra feels its way towards an interpretation. The role of first and second violin keeps changing between orchestra members and in the course of a rehearsal players from different sections might swap desks to hear what sounds best. It's all done good-humouredly — but it's also a labour of love, with participants performing without pay in Spira's first year.

The ensemble meets up about seven times a year and rehearses in different towns up to 10 hours a day for the four or five days leading up to a final concert, attracting interest through the spontaneous appearances they make in supermarkets, malls and squares. Fregni talks proudly of playing to audiences of ordinary people, "policemen, shop keepers and pensioners".

Outside a rock club in Hamburg's red-light district last month, I was among the crowd who queued for a concert by Spira followed by experimental DJs. Inside, the music stands were set up not on the stage but in the body of the hall, with the audience standing around them.

Dressed in black, the players shook hands with the audience as we arrived and Fregni talked with gentle passion about the piece we had come to hear, before the orchestra struck up with masterful fluency and honest intensity.

It was magnificent. Londoners at the QEH — and the unsuspecting members of the public who will be treated to a guerilla performance somewhere in the capital this week — are in for no less a treat.

Spira Mirabilis perform as part of Shell Classic International at the Queen Elizabeth Hall (0844 847 9910; southbankcentre.co.uk) on Friday.