The most notable conservative dissonance on policy during the Sino-Soviet split was a widespread belief that the "split" was a fraud. This argument had strong partisans in the CIA, notably counterintelligence chief James Jesus Angleton; from that center of skepticism various other right wing media picked up the argument. James Burnham, publisher of the National Review with William F. Buckley, would be a pretty typical example of the "its a fraud" argument.
Different elements of American communism sided variously with the Soviets and Mao. Very generally, the Hitler-Stalin Pact and then Krushchev's "Secret Speech" tended to diminish whatever enthusiasm American leftists might have originally felt for Soviet communism. A pro-Moscow American Communist Party persisted, but was an afterthought.
So in the 1960s Mao was newer and more interesting to some leftists. You get a bit of buzz around the Cultural Revolution, some leftists fancying themselves Maoists. Mao's "Little Red Book" had a degree of currency in the West that no equivalent Soviet propaganda did . . .
In the West, Maoist sympathies were perhaps strongest in France, where it had both an exotic and an intellectual/revolutionary appeal that the by-now-very-grey Soviets had lost.
Sources
Four Theories in Search of a Reality: James Burnham, Soviet Communism, and the Cold War
How the west embraced Chairman Mao’s Little Red Book
https://www.newstatesman.com/culture/2014/05/how-west-embraced-chairman-mao-s-little-red-book
How Maoism Won the West: But What Exactly Did Mao and Malraux Say to Each Other?