Evidence is found in the history of the founder of the British Union of Fascists - Oswald Mosely - who was a Labor Parliament Member of Parliament and a committed socialist before founding the British Union of Fascists:
Mosley was at this time falling out with the Conservatives over Irish policy, objecting to the use of the Black and Tans to suppress the Irish population. Eventually he 'crossed the floor' and sat as an Independent MP on the opposition side of the House of Commons. Having built up a following in his constituency, he retained it against a Conservative challenge in the 1922 and 1923 general elections.
The liberal Westminster Gazette wrote that he was "the most polished literary speaker in the Commons, words flow from him in graceful epigrammatic phrases that have a sting in them for the government and the conservatives. To listen to him is an education in the English language, also in the art of delicate but deadly repartee. He has human sympathies, courage and brains."[5] By 1924 he was growing increasingly attracted to the Labour Party, which had just formed a government, and in March he joined. He immediately joined the Independent Labour Party (ILP) as well and allied himself with the left.
When the government fell in October, Mosley had to choose a new seat as he believed that Harrow would not re-elect him as a Labour candidate. He therefore decided to oppose Neville Chamberlain in Birmingham Ladywood. An energetic campaign led to a knife-edge result but Mosley was defeated by 77 votes. His period outside Parliament was used to develop a new economic policy for the ILP, which eventually became known as the Birmingham Proposals; they continued to form the basis of Mosley's economics until the end of his political career.
In 1926, the Labour-held seat of Smethwick fell vacant and Mosley returned to Parliament after winning the resulting by-election on 21 December. Mosley felt the campaign was dominated by Conservative attacks on him for being too rich and claims he was covering up his wealth.[6]
Mosley and his wife Cynthia were committed Fabians in the 1920s and at the start of the 1930s. Mosley appears in a list of names of Fabians from Fabian News and Fabian Society Annual Report 1929–31. He was Kingsway Hall lecturer in 1924 and Livingstone Hall lecturer in 1931.
Odd, I've never heard of Mosley's Labor Party involvement.