This is a great question that could not have been asked at a worst (or better) time. Twenty years ago, I would've told you that Chinese culture was profoundly changed in daily life in the brief but chaotic turmoil of the Cultural Revolution; a total 180 of cultural values in which many Chinese youth, utterly dedicated to Mao's cult of personality, sought out to destroy the Four Olds and then immediately calmed down back into society after Mao’s death. Or at least, this was generally an accepted consensus by historians in the 1980-2000s, and this is probably what most Westerners believe when they think of the time period. And there is still some truth in this analysis, but recent research in the 2010s has shown that this is not holistically true. Barbara Mittler's A Continuous Revolution: Making Sense of Cultural Revolution Culture (2012) has changed the research landscape of this very topic.
Mittler's main argument in this book was to fight against the idea that Chinese culture during the CR was stagnant and full of meaningless propaganda surrounding Mao. Rather, she argues that the cultural output (mainly through propaganda, music, and literature) flourished into rich diversity. She goes even further, arguing that Chinese culture at this time did not represent a total shift toward xenophobia (toward Westerners) and iconoclasm (toward tradition).
Music
One would assume that China's massive output of anti-Imperialist propaganda at this time would lead to most Chinese destroying or forgetting Western pieces of art, literature, and/or music. We have much evidence however, mostly through propaganda posters and calendar images, that while the CCP did have strong regulations on what could and could not be shown/heard, many commoners continued to practice and enjoy things that they were introduced to since the late 19th century. A calendar poster dated 1975 shows two images, both of girls, playing two different instruments. The work is titled "Get rid of what is stale and bring out the new," and shows one (older) girl playing a guqin, a Chinese stringed instrument, and another (younger) girl playing a violin (an instrument not familiar to the Chinese until roughly the late 19th century). Government regulation also intervened:
Music, just like all other artistic production, was subject to extreme political regulation during the Cultural Revolution; only certain correct colors, forms, and sounds were officially acceptable: Wolfgang Amadeus Mozart (1756–91), Ludwig van Beethoven, Franz Schubert, and Johannes Brahms (1833–97) were condemned due to their “bourgeois” background or upbringing; Arnold Schönberg (1874–1951) and Claude Debussy (1862–1918) were considered “formalists”; Pyotr Ilyich Tchaikovsky and Sergei Rachmaninoff (1873–1943) were said to be (pre-) representatives of the “revisionist” Soviet regime and thus could not be performed; sounds of the guqin were unacceptable, as they were associated with the “aristocratic” literati of “feudal” China; and traditional Chinese operas were said to bring too many “emperors and ladies,” and too few “workers, peasants, and soldiers” onto the stage. And this list could be continued.
There is a lot of blending of Chinese and Western culture in music. It's very complicated for the CCP because while Westerners are an enemy, communism is ultimately a Western ideology. And while Chinese cultural-variants of communism are good (Maoism), there are cultural aspects of China that are bourgeoisie.
Music during this time saw a mass amount of urbanite composers entering the mainstream composing operas and songs dedicated to "Serving the People." Operas have for quite some time been an important cultural aspect of China, particularly in Beijing. Mittler argues that these operas redeveloped traditional opera styles rather than destroying it. And for songs, we know well that the Internationale was sung regularly by workers, and Western music was so assimilated into China that by the 1970s, the song "Frere Jacques" became a battle song of some political factions. From interviews with those who lived through the CR, we learn that accordions and guitars were quite popular beginning in the mid-1960s, and that the music of choice to record/play were Russian.
So in conclusion for music, we learn that underneath all the politics, the common Chinese citizen still enjoyed deep interests in Western/traditional music. The major change in theme during this time period, however, is the relation of music and opera towards two things: 1) The People and servitude towards them, and 2) the hard, cold struggle of the Chinese worker. By the 1980s however, these themes mostly disappear as China's younger generations experience economic stability and success their parents and grandparents never did.
Literature
Literature in China during the CR actually flourished in many different ways than was originally thought. The best example, used by Mittler, is the re-purposing of the Sanzijing (Three Character Classic) for purposes of serving the Worker's Revolution:
During the “Campaign to Criticize Lin Biao and Confucius,” in short, the Anti-Confucius Campaign, which took place from 1973–75, the Museum of Revolutionary History in Beijing published a “Selection of Three Character Classics for the Peasants, Workers, and Soldiers of the New Democratic Revolutionary Age” (新民主主义革命时期工农兵三字经选 Xin minzhu zhuyi geming shiqi gongnongbing Sanzijing xuan) (XMZZY 1975), which brings together different renditions of the Three Character Classic which appeared between the 1930s and the 1940s. The collection emphasizes in its preface that these “revolutionary” versions of the Three Character Classic are all the result of a struggle against the hateful ideas of the original Three Character Classic (革命三字经就是这种斗争的产物) (Ibid., 4). They are considered renditions of the classic work that, by taking on the very form of the original, are able to fight the ideas contained therein ever more effectively.
This re-purposing of very old scripture to conform to modern Maoist ideology is a big example of Chinese literati at this time still hanging on to traditional cultural aspects in order to answer modern questions. Scripture wasn't the only thing re-purposed during this period; Mao was very fond of using Chinese folk stories and metaphors in order to represent class struggle. His favorite was "The Foolish Old Man and the Mountain," an old Chinese fable about an old man, who, using a bucket and a hoe, attempts to get rid of the mountains which obstructed many of his duties. Long story short, when told that he is a fool, he replies that through the hard work of his own labor, and his future generations, the mountains will eventually be moved, and then the Gods move them for him as they are impressed with his dedication. The story served for Mao as a perfect parallel between the struggle of the Chinese peasant and the bourgeoisie in the Little Red Book. In fact, the Little Red Book is essentially to Maoists at this time what the Analects were to Confucians in ancient times. Mao effectively re-purposed traditional stories and kind of just turned many Confucian rituals into Maoist ones.
In conclusion for literature, many intellectuals in the CR re-purposed lots of traditional Chinese works into fables and moral stories for the struggle of the Chinese worker. This is symbolized in the Little Red Book, which Mittler mentions every Chinese citizen was able to (and probably did) own a copy (there are hundreds of millions of Chinese LRB copies).