But new evidence has emerged recently showing that the Board, in fact, was doing exactly what one would have hoped, and more, by infiltrating Sir Oswald Mosley's British Union of Fascists. The Board also, apparently, paid members of the Communist Party to take part in the street protests, in which they stood alongside trade unionists, Irish dockers - and, most fundamentally, the Jews of the East End.
Mr Sugarman would like to see a different emphasis on the commemoration of Cable Street. "We don't know who was or wasn't there," he says, "but people do embellish. Now we have a situation where those who want to show that they are not antisemitic speak about Cable Street, to give themselves credibility. And left-wing Jews have bought into the myth."
Clive Bettington, a lawyer who is chairman and founder of the Jewish East End Commemoration Society (JEECS), is cynical about those who claim to have been at Cable Street, noting sharply that "no respectable parent would have taken a child to a place where there would be almost certain violence".
He is keen to bust the other main myth about Cable Street, that of direct confrontation between the fascists and the Jews. There was violence, but it was between the police and the East Enders as the police formed a barrier between Mosley's blackshirts and residents.
"People wonder why Mosley turned round so quickly," says Mr Bettington. "Partly it was because it suited the fascists to portray themselves as victims, but it was also because two days after Cable Street, Mosley was due to marry Diana Mitford in Berlin, in Josef Goebbels' home, with Hitler in attendance. The last place Mosley wanted to be was in a UK police cell for not complying with police orders."
The most tangible legacy of the Battle of Cable Street was the 1936 Public Order Act, although opinions differ as to who or what was the Act's architect. David Rosenberg, of the Jewish Socialists' Group, gives credit to the 1930s body, the Jewish People's Council Against Fascism and Antisemitism, which had called for a law against racial incitement.
For Mr Rosenberg, "the fundamental message of Cable Street is that of how solidarity and collective action can bring communities together and prevent fascism, and of being brave and courageous in the face of threats from the far right."
The left, he believes, "has a very legitimate claim to be the heirs of Cable Street, inside and outside the Jewish community."
But he acknowledges that the way in which it is commemorated by the mainstream Jewish community has changed, from being "a bit reticent" to a "full-on desire to embrace a victory".
Daniel Tilles, assistant professor of history at the Pedagogical University of Krakow, is one of the pre-eminent researchers into Cable Street. For him, mainstream Jewry's slow change of heart in its attitude to Cable Street is "a sign of greater Jewish confidence about its place in society".
He says: "It was a controversial event at the time - 88 people were arrested, and there was some reluctance to claim ownership. In fact, fascist membership increased after the event and life was worse for East End Jews because the fascists used Cable Street to feed into a narrative that they were being victimised by Jews."
Dr Tilles says his research shows that the Board "did a huge amount behind the scenes, paying agents to infiltrate the fascists and giving information both to the Home Office and the police," via "weekly meetings" with the Board president, Neville Laski.
Cable Street "absolutely must" be commemorated, says Dr Tilles. "The lesson of the '30s is that far more can be achieved by co-ordination among different groups and lobbying authorities."
On the other hand, he says, "it allows anti-fascist groups to burnish their credentials by giving publicity to the far right. It is a difficult balance."
The graphic designer and photographer Simon Wilder is the son of a Holocaust survivor and the grandson of Hymie Leigh, who was at Cable Street in 1936. Mr Wilder says: "My grandfather fought at Cable Street. Actual fighting. His nose was broken. We were aware of it, almost as much as we were about my father being in concentration camps. It was a defining moment for my grandfather, as well, I think, as for other London Jews and anti-fascists".
But he is scathing about the way in which Labour leader Jeremy Corbyn "wraps himself in the flag" of Cable Street. He says: "Jeremy Corbyn wasn't there. He hadn't even been born. If - if! - his mother was there, it does not mitigate his position on Jews and Israel now. There isn't a pair of moral scales - 'if my mother did this, it clears me of that'. How dare Corbyn claim Cable Street for his moral armour."
Mr Corbyn, who makes a point of talking about his mother's participation at Cable Street whenever he is in front of a Jewish audience, will be taking part in the cross-community march and rally on Sunday, October 9.
The London Jewish Forum, which, together with the Jewish Council for Racial Equality and the Jewish Socialists Group, is a supporting organisation of the march and rally, is holding a separate commemoration later on the same day.
Together with the Jewish Leadership Council, the LJF has assembled a hand-picked line-up - the Mayor of London, Sadiq Khan, Chief Rabbi Ephraim Mirvis, Rabbi Laura Janner-Klausner, MP Ruth Smeeth, and the general secretary of the TUC, Frances O'Grady.
Adrian Cohen, chair of the LJF, says: "The Jewish response to fascism at Cable Street led to the creation first of the 43 Group and then, eventually, to the launching of the Community Security Trust, the CST.
"There is a benefit in remembering events for their own sake, and it would be a great shame if the new generation didn't know about Cable Street. People growing up in middle-class environments shouldn't forget what life was like for those Jews in working class areas.
"But there is a danger of being overly nostalgic, and losing sight of the contemporary situation. Fear of Jews mutates, and if Cable Street teaches us anything, it is to remain aware."
The CST founder and chair, Gerald Ronson, says: "I have spent decades leading the communal fight against antisemitism, and I see that as continuing the proud tradition that is marked by Cable Street.
"Over time, our community has developed, but so have the threats. Things may have changed, but at heart it remains the same fight as it has always been."
