I think this is a VERY interest question that gets to the heart of what logical positivism, and empiricism in general, are really about.
I remember Peter Hacker in his book 'Wittgenstein's Place in 20th Century Philosophy' explaining the Marxist leanings of some (but not all) of the leading positivists. Logical positivism was a epistemic reductive movement with a tendency to elevate the status of science culturally and strongly reject metaphysics/religion. In this respect logical positivism broadly belongs to empiricism. Ever since Hume, empiricism has had a strong connection with atheism, the rejection of religion, and the promotion of science in its place. Logical positivists tended to continue in this tradition. To some positivists, Marxism probably seemed more compatible with their empiricist dogmas than the free market based political systems that most Western democracies had implemented in the early 20th century. There was certainly a strong cultural match between the progressivism towards science inherent in logical positivism and the cultural progressivism of Marxist idealology. The positivist manifesto seems to be influenced by the earlier Communist Manifesto, for example.
I think it is reasonable to say that to this day there persists a general cultural connection between scientific materialism (the successor to positivist scientism), atheism, and Marxist/socialist ideologies. But this is my own humble opinion which needs further argument.