Last weekend, Tony Brett, a Liberal Democrat councillor in Oxford and the city’s deputy lord mayor, found what he called a “disgraceful rabble” of people climbing on the city’s main war memorial — squashing, he said, the flowers that mourners had placed there, then trying to remove half of them altogether and “jeering” other visitors as they paid their respects.

That day, the memorial was supposed to be the scene of a wreath-laying by the far-Right, racist English Defence League. But none of the people laying flowers and being jeered bore any kind of EDL insignia and none of the wreaths had any kind of card or message from the group.

Neither Mr Brett, nor a local newspaper reporter on the scene, saw any sign of any EDL presence.

All the aggro, Mr Brett said — he called it the “hate” — came from the self-appointed opponents of bigotry, a group called Unite Against Fascism (UAF). UAF’s response was to start an online petition saying that merely by criticising them Mr Brett had proved himself an EDL patsy, “not a fit representative for Oxford’s wonderful and multi-ethnic community”, and must resign immediately.

“It seemed to me they were doing exactly the kind of thing they were supposed to be protesting against,” said Mr Brett. “I will absolutely not support any hint of racism, Islamophobia or any other form of hate, be it from the EDL or any other group. That day I saw it from another group.”

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The Oxford branch of UAF said its members climbed on the memorial at the request of a photographer. “The EDL’s use of war memorials is an offence to all those who died fighting fascism,” it said in a statement. “Mr Brett has given support to an event designed to boost the credibility of a fascist organisation.”

UAF, 10 years old this year, is one of Britain’s most prominent anti-fascist organisations. It has received hundreds of thousands of pounds from the biggest trade unions, and support from dozens of mainstream politicians. Its vice-chairmen include Christine Blower, the general secretary of the National Union of Teachers, and Hugh Lanning, the deputy general secretary of the PCS civil service union.

Even David Cameron, when a backbench MP, signed its founding statement. It has avoided substantial scrutiny because, as in the case of Mr Brett, anyone who criticises it risks being smeared as an ally of the vile groups it opposes.

Of course, few causes can be more deserving than resistance to the EDL and British National Party. But the uncomfortable truth about UAF is that it contains more than a trace of fascism itself. It specialises, as seen in Oxford, in organising counter-demonstrations to any activity, or anticipated activity, by the far Right.

Unfortunately, UAF’s counter-demonstrations often seem to cause as much, if not more, trouble than those by the EDL and BNP.

Since the murder of Drummer Lee Rigby last month, there have reportedly been at least 107 arrests during BNP, EDL and UAF demonstrations. At least 69 of those arrested, just under two thirds, were anti-fascist demonstrators, at least 58 of them UAF.

Weyman Bennett, UAF’s secretary, said none of the 58, all arrested on an anti-BNP counter-demo in Westminster two weeks ago, had so far been charged. “They haven’t done anything,” he said.

The arrests came after UAF demonstrators tried to prevent BNP supporters from reaching the Cenotaph and refused to stay in their allocated protest space. Video footage showed a BNP supporter being attacked by some UAF demonstrators, suffering cuts to his face.

“Our view is more complicated than saying we just want confrontation,” said Mr Bennett. “When racists organise, you can’t wait until they become a large and unstoppable force. You have to show there’s more of you than there are of them and you have to show they’re confronted by people of different races holding hands.”

In 2010 UAF’s then national officer, Martin Smith, was given a 12-month community order after being convicted of assaulting a police officer on a demonstration. He remained in his post, insisting his conviction was an “outrageous attack on the right to protest against fascists”.

A senior UAF official has recently been accused of rape and sexual assault by two women from the Socialist Workers’ Party, in which he was also an official. The man, who cannot be named for legal reasons, has given up both posts but continues to take part in UAF protests and activities, including as recently as last week.

Prominent campaigners such as the gay rights activist Peter Tatchell accuse UAF of a selective approach to bigotry. “UAF commendably opposes the BNP and EDL but it is silent about Islamist fascists who promote anti-Semitism, homophobia, sexism and sectarian attacks on non-extremist Muslims,” said Mr Tatchell. “It is time the UAF campaigned against the Islamist far Right as well as against the EDL and BNP far Right.”

One reason why UAF will not campaign against Islamist extremists is that one of its own vice-chairmen, Azad Ali, is one. As well as his UAF role, which he took up last year, Mr Ali is community affairs coordinator of the Islamic Forum of Europe, a Muslim supremacist group dedicated to changing “the very infrastructure of society, its institutions, its culture, its political order and its creed from ignorance to Islam”.

Mr Ali has written on his blog of his “love” for Anwar al-Awlaki, the al-Qaeda cleric closely linked to many terrorist plots, including the September 11 attacks, and used to attend talks by Abu Qatada, the extremist cleric whom Britain is seeking to deport.

He has described al-Qaeda as a “myth” and denied that the Mumbai attacks were terrorism. On his blog, he also advocated the killing of British troops in Iraq. He sued a newspaper for reporting that he had said this, and lost.

Filmed by an undercover reporter for The Sunday Telegraph and Channel 4’s Dispatches, he said: “Democracy, if it means at the expense of not implementing the sharia, of course no one agrees with that.” Mr Bennett defended Mr Ali, saying: “He’s done valuable work for us. I’ve heard him speak on many occasions and he’s never said any of the things he’s been accused of.”

One of the alleged killers of Drummer Rigby, Michael Adebolajo, also spoke on the margins of a UAF rally in Harrow in 2009. Video footage shows him addressing a crowd at the event. Mr Bennett said that he was not on the platform, nor was he an official speaker.

There are effective ways to defeat the far Right — the group Searchlight has inflicted significant damage through voter-registration drives, door-to-door campaigning and precise, well-researched intelligence driving a dagger into its real weaknesses.

And there are ineffective ways. The racist Right thrives on two things: publicity and the politics of victimhood. The mob outrage practised by UAF gets the fascists more of both. As with the “anti-Islamophobia” monitoring group Tell Mama, which has lost its government funding after overhyping the nature of anti-Muslim hostility, there is a sense that the racists and their opponents need each other.

Mr Brett added: “It just antagonises the situation. The way to deal with this stuff is not to fight it aggressively. That’s exactly what they want you to do.”

Nobody has denied that there has been an increase in tensions since the murder of Drummer Rigby.

The danger is that by exaggerating it, and by the politics of confrontation, supposedly anti-racist groups fuel the very division, polarisation and tension they are supposed to counter.