I went and saw the mural of the Battle of Cable Street at Stepney Borough Council and the plaque put forth these claims. I'd like to know more about the strength of the fascist movement at the time and the impact this "battle" had on it.
It's a wonderful mural, although to my (very specifically prejudiced) mind there should be better representation of the ongoing war in Spain within the mural itself. I believe there is a memorial plaque to local volunteers who went to Spain around the corner, at least...
In any case, I answered a very similar question on Cable Street a month or so back. That's certainly not to say that there isn't anything more to say on the subject, and I'm happy to expand on points or answer any follow up questions as best I can.