What happened at the Battle of Cable Street?

The death on 2 November at the age of 103 of veteran Stalinist Max Levitas has revived attention on this anti-fascist event of 4 October 1936 in which he had played a part. According to his obituary in the Times (16 November):

Sir Oswald Mosley, the leader of the British Union of Fascists, had announced that he was planning a march of about 5,000 of his Blackshirts through an area of the East End of London populated predominantly by Jews (...)
The Jewish People's Council Against Fascism and Anti-Semitism sprang into action, collecting a petition of 100,000 names calling on the home secretary, John Simon, to ban the march. Simon refused, but a secret arrangement was made whereby Mosley-would re-route his march on the day at the "request" of the police commissioner and there­by score a propaganda victory. His Blackshirts had been getting a repu­tation for disorderly con­duct and this would prove they were law-abiding.
The Blackshirts assem­bled in Royal Mint Street and Mosley arrived to inspect them in his open-topped Bentley. They were then, as arranged, escorted to Embankment and ordered to disperse.
Half a mile away, about 30,000 anti­fascists were massing behind barri­cades along Cable Street. When the mounted police charged them, the situation escalated quickly, with anti­fascists ripping up cobbles from the streets and throwing them, along with bottles and rotten vegetables. The ensuing riot lasted for three hours and resulted in hundreds of arrests.
Levitas, who served as one of the Communist Party's messengers that day, recalled how anti-fascists threw marbles on to the ground to make police horses slip. "Mosley and his fascists wanted to take over the East Ivnd," he said. "To run out the Jews and communists. We had to stop them. It was the people united, fighting to­gether. When we heard that the march was disbanded, there was a hue and cry. The flags were going up wildly — the red flags and others — and people shouting 'We have won!'"
To the extent that it looked as if the Blackshirts had run away from a fight with "the reds", it did indeed seem as if the anti-fascists had won.

If this account is accurate, then the battle of Cable Street was not a battle to stop the Mosleyites marching on the East End with police protection (as there was a secret deal with the police not to do this in the end) but a battle against the police who were trying to contain the anti-fascists assembled in Cable Street. Of course in any event the mobilisation of people in Cable Street contributed to Mosley backing down, but if this account is accurate it was not theanti-fascists who physically prevented the Mosleyites from marching.
But is the account in the Times obituary accurate?