It's worth noting that "cultural Marxism" is not the same as actual Marxism.
Cultural Marxism is a term used primarily by right-wingers to refer to progressive approaches to culture, particularly anything that suggests that culture is a fundamental cause of inequality. The relevance to Marxism implied in the name is part conspiracy theory and part slur; basically the claim that such ideas are fundamentally derived from Marxist ideology (and therefore the well is poisoned or something). It's generally identified with the
Frankfurt School when they bother to actually mention any real academic tradition rather than just throw the phrase around
Nobody actually calls
themselves a cultural Marxist, as far as I know. It's more of an accusation to use against people you dislike than an ideology.