The spectacle of hundreds of Lazio fans doing straight-arm, Fascist salutes as they marched through the streets of Glasgow on Thursday was repugnant, but hardly surprising.

In Italy, hardcore football fans are called ultras, and Lazio’s ultras are the most hardcore in the country: called the Irriducibili (the “irreducibles” or “die-hards”), they are infamous not just for their political extremism but also for bank-jobs, drug-dealing and unsolved murders. Far from being football fans, the Irriducibili are a criminal gang.

It was the city’s misfortune that the match between Celtic and Lazio was on October 24: the day before, the 23rd, marked the anniversary of the Budapest uprising of 1956, always celebrated by the Irriducibili as a symbol of underdogs rebelling against Communism.

As they goose-stepped through Glasgow, the Irriducibili were singing their anthem, Avanti Ragazzi di Buda (“come on kids of Buda”). The timing was also sensitive because, just last week, UEFA decided that Lazio’s Curva Nord (the “north terrace”, which is home to the Irriducibili) would be closed for Celtic’s away leg in Rome on November 7 because of previous Fascist salutes in their match against Rennes. So Thursday was the last chance for the defiant extremists to strut their stuff on the international stage.

But there are deeper sub-plots at work, too. In August, the Irriducibili’s leader, Fabrizio Piscitelli (nicknamed “Diabolik”) was murdered as he sat on a park bench in Rome. It was a professional hit (a bullet to his left ear from someone dressed as a jogger) and had all the hallmarks of an underworld reprisal. Piscitelli had previously been convicted of handling hundreds of kilos of narcotics, and had recently had assets of over 2m euro seized.

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There is now a power-vacuum at the top, and because the Irriducibili are leaderless, various lieutenants are jostling to become top-dog. Each aspiring leader is trying to prove they’re more Fascist, more “ultra”, than their rivals. There’s also external pressure: the Irriducibili have bossed the Lazio terraces for almost three decades now, and in recent years various rival groups, like the “Hit Firm”, have attempted to take their crown. At this time of possible vulnerability – leaderless and under attack from football officialdom and the Italian police – the Irriducibili inevitably wanted a show of force, all the better if abroad.

Sadly, there’s nothing new in all this. In the early 1970s, at the birth of the ultra movement, many Laziali toyed with Fascist insignia – celtic crosses, double-headed axes, even swastikas. It was, as with the Hells Angels and punks, often just a way to spook straight society. But over the years, Lazio became a haven for true believers nostalgic for Benito Mussolini’s totalitaranism: they organised formal twinnings with other Fascist groups from Inter and Verona, and many black players – like Lillian Thuram – refused offers to join the club.

It would be easy to dismiss the Irriducibili as mindless thugs, but – unlike British hooligans of old – the Irriducibili are highly organised and calculating. It was noticeable on Thursday that they were almost all dressed identically, in blue jeans and black jackets. The group has always appeared hierarchical and paramilitary: it was the first in Italy to erect huge speakers in the terraces so that one leader, with one microphone, could dictate all the singing. The group set up a multi-million merchandising operation, under the “Original Fans” label. At one point they had 14 outlets. Their strength was such that, in 2005, they even provided the muscle, and threats, for a hostile takeover bid of Lazio using Mafia money and a club legend, Giorgio Chinaglia, as an unwitting front man.

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Since then, the Irriducibili have constantly been in the news for their provocative stunts and anti-Semitism. But the most serious aspect of the ultras’ extremism is the fact that it is indulged and excused by so many within Italian society. It has become fashionable to express admiration for Mussolini here, and far-right political parties – the “League”, “Brothers of Italy” and all the others – are currently, combined, polling well over 30 per cent. Of course, millions of Italians and many ordinary Lazio fans are appalled by it all, but sadly what was shocking for Scottish citizens is a grim, weekly occurrence in the Italian capital.

Tobias Jones lives in Parma, Italy. He is the author of Ultra: The Underworld of Italian Football.