It is a mistake to confuse 'classical' Marxism with cultural Marxism.
The former is characterized by the labor theory of economic value; the call for the abolition of private property; collective ownership of the means of production, i.e., socialism in the strict sense of the term; historical materialism (HISTOMAT) and dialectical materialism (DIAMAT); belief in objective truth (see V. I. Lenin); the Hegel-inspired belief that history is being driven in a definite direction by an in-built nisus towards a secular eschaton*, in the case of Marx & Co., the dictatorship of the proletariat and the classless society . . . You know the drill.
But as Paul Gottfried points out, cultural Marxism is a horse of a different color. In particular, it is not usefully or reasonably labelled socialist. Gottfried's insights (in this article) need to be taken on board, not that I agree with everything the man says elsewhere.
*A really deep understanding of secular eschatology such as we find it in Marx requires a critical retrieval of Christian eschatology. Please forgive my 'critical retrieval.' Back in old Boston town, in the early-to-mid-seventies, I was a bit of a Continental philosopher. I sipped a little of the Leftist Kool-Aid, but never got drunk on it, despite all the Habermas, Horkheimer, and Adorno I read. Gott sei dank!
Perhaps I can thank Heidegger for saving me. My intense occupation with his writings and his Seinsfrage drove me back to Aquinas for the onto-theological approach to Being and to Frege and the boys for the logical approach.