Matteo Salvini, Italy’s far-right deputy prime minister, this week announced the creation of an alliance of radical rightwing parties called “Toward a Europe of common sense”, which will attempt to become the largest group in the European parliament after next month’s EU elections. Frightening stuff. But how genuine is the threat?

Could a new quasi-fascist axis be forming in Europe? In his 2004 book, The Anatomy of Fascism, the American historian Robert Paxton wrote about the importance of distinguishing “ugly but isolated imitations [of fascism], with their shaved heads and swastika tattoos, from authentic functional equivalents in the form of a mature fascist-conservative alliance”.

The far right has a chance of succeeding if it finds allies beyond its own, strictly partisan realm

Trips in recent months to northern Italy, Budapest, Warsaw, provincial France, Berlin and Stockholm have convinced me that Paxton’s 2004 writings help us clear up much of the fog that comes from either exaggerating the clout Salvini and his “alliance” can garner in the EU, or unduly minimising their influence.

The crucial thing is not so much how many votes far-right parties will by themselves pick up across the EU in six weeks’ time (although, of course, that’s a worry), but whether those forces will seek, and be able to form, “a mature fascist-conservative alliance”, alongside what Paxton called “threatened conservatives looking for tougher allies”.

Polls show Salvini is currently far off the mark: far-right parties put together are set to win an estimated 150 seats out of a total of 705 in the next European parliament. That’s far from a majority, and far from the number one position (the mainstream rightwing European People’s party is in the lead, with a projected 180 seats). When Salvini announced his “alliance” in Milan on Monday only three other parties shared the platform: the German AfD, the Finns party and the Danish People’s party. Take note of the absentees: no Marine Le Pen (France), no Viktor Orbán (Hungary), no Heinz-Christian Strache (Austria), no Jimmie Åkesson (Sweden). That made for an unimpressive start.

Drawing comparisons with the 1920s and 30s has become commonplace: many on the left, as well as many of those who recite the “progressives versus nationalists” credo (think Macron), often indulge in it. This is understandable: what better way to mobilise voters than to say Europe is in danger of falling prey to overtly xenophobic, intolerant, propaganda-fuelled, strongman-loving parties. Shouting “fascism!” is an effective clarion cry, though it can sometimes tip into self-gratifying virtue signalling.

At the other end of the scale there are those who minimise the scale of the threat, depicting Europe’s far right as chronically disunited,; a hodge-podge of diverging interests and sensitivities. How can these parties be both international-minded and nationalist? Or both love and hate Putin ? Moreover, they argue that immigration isn’t as big a deal to European voters as Salvini and company like to think it is. Corruption, inequality, and even emigration (in southern Europe) are of bigger concern, surveys show.

The truth lies somewhere between these two judgments. The next EU parliament will be more fragmented than the current one – large traditional groupings will wane and new or insurgent ones will rise. The redrawing of domestic political maps will naturally translate into that European assembly. But, again, far-right parties are unlikely to resemble a tsunami sweeping everything before it.

What will undoubtedly come into sharp focus is the meandering of social democracy; Pasokification” (from the Greek example), the struggle of traditional left-of-centre forces to find a potent message in the age of middle-class disgruntlement. The greens might fill some of that void. And Macron’s En Marche!, with its “renaissance” slogan, will make its entry. A complex picture will emerge, not a one-sided land grab of would-be dictators. What will matter most is what comes after the election. And this is where Paxton’s warning comes in. He argues: “Fascists are close to power when conservatives begin to borrow their techniques, appeal to their mobilising passions, and try to co-opt the fascist following”. While in Budapest last month, I was horrified to discover that Nicolas Sarkozy, the former French president, had embraced Orbán on stage, just a few days after Hungary’s self-proclaimed “illiberal” leader had been suspended from the European People’s party. It was yet one more sign that France’s mainstream right is courting hardliners and extremists.

Orbán and Salvini – the man who likes to quote Mussolini – already have a record of teaming up together, and Orbán has methodically dismantled democratic checks and balances in Hungary. None of this has yet convinced Europe’s mainstream right that he should be kicked out of its tent in the EU parliament entirely (his Fidesz party is only suspended). As Paxton suggested while writing about fascists in Europe’s interwar period, that kind of approach can only serve to legitimise both Orbán and Salvini’s rhetoric, creating yet more ground for “democracy gone wrong”, as the historian put it.

The far right has a chance of succeeding in its culture and identity wars if it finds allies beyond its own, strictly partisan realm. As long as fascists remain “excluded from the alliances with the establishment necessary to join the political mainstream or share power” they are a limited threat, wrote Paxton. That type of alliance has already taken shape in some EU countries – Austria, for instance. The question is: will it happen in the next EU parliament?

Far-right parties won’t have the capacity to take control of the EU alone, but their ideas have undoubtedly contaminated others. The onus will be on the mainstream right to prove there’s enough political will, and enough historical awareness, to prevent nationalist and racist demagoguery from getting near the bloc’s levers of power. That’s where the key test lies. For Manfred Weber (the leader of the European People’s party), Paxton should be a must-read.

• Natalie Nougayrède is a Guardian columnist