There is trinitarian aspect to Communism, with Marx being the Father, Lenin the Son, and Gramsci the Holy Spirit (or, by some lights, the "anti" versions of the Father, Son, and Holy Spirit).
"Cultural marxism" arose after the failure of Marx' economic predictions, and the repression of Lenin's governance, as a kind of "Hail Mary!" effort to overthrow the Christian West (i.e., after the economic theories had proven wrong, and the political behavior had proven oppressive). The "third way" was Gramsci's approach to defeating, overturning, the West. See this for a sympathetic portrayal of the general strategy:
http://en.wikipedia.org/wiki/Antonio_Gramsci
"Cultural marxism" is to be distinguished from "political correctness," which originated as an ironic phrase in U.S. CPUSA circles, i.e., for the Communist when dealing with the non-Communist, "pc" was to appear to be non-Communist; and, of course, to actually be Communist, the other side of the "pc" coin. Cultural marxism was first majorly employed by Soviet Communists in the Khrushchev era, when Gramsci's strategy was developed as the number 2 salient in Soviet aggression against the West, via the development of drug cartels in South America, training the leaders of such in Czechoslovakia and the Soviet Union, bribing, blackmailing, and intimidating South American leaders to "accept" the drug cartels, and then promoting the introduction of cheap heroin and other drugs into U.S. inner cities, among the youth culture, and in U.S. military stations. (See "Red Cocaine," which documents some of this; also, U. S. PoW and Russian prisoners were used to test new drugs.) (The first salient of Soviet aggression continued to be a nuclear first-strike plan, however, the Soviets were daunted by the U.S. retaliatory capability at the time; a third salient was the development of biochemical weapons of mass destruction (see "Biohazard: The Chilling True Story of the Largest Covert Biological Weapons Program in the World -- Told from Inside by the Man Who Ran It," Alibek and Handelman;
http://en.wikipedia.org/wiki/Ken_Alibek ). A fourth salient was the general spy and military activity (see "Inside the Aquarium: The Making of a Top Soviet Spy," by Viktor Suvorov, and "The Venona Secrets: The Definitive Expose of Soviet Espionage inn America," Romerstein and Breindel) of the time.
Over time, "cultural marxism" has become the most successful tactic of the four (although recent cyberwarfare is close:
http://www.schneier.com/crypto-gram/archives/2016/0915.html#2 ).
Related: "Tavistock Institute" by Daniel Estulin, and "The Soulless One" by Mark Prophet.