Some research done in the 1960s and 1970s is relevant to this question. They found remarkable synchronies and symmetries among body movements and speech sounds, and found many of these present already in the interaction between mother and infant, which Stern likened to a waltz. He and others proposed that in autism this dance was disrupted, and Condon had film to show that it was. The cerebellum would have to be involved in coordinating this dance, and its role would be a natural extension of its role in coordination of movement across joints and in integration of composite movements, by adding information about movements of the other to the information it has about movements of the self. Significantly, vermal lobules VI and VII and related paravermal regions are the parts of the cerebellum to which the auditory and visual senses connect24, the senses the cerebellum would have to use to play this additional role.
Autism is more than just the autistic triad.
As Temple Grandin25 and other people with autism eloquently point out, the disorder includes significant sensory abnormalities as well. These used to be a focus of autism research, but this research also fell out of fashion. Any theory of autism has to explain them, however. Interestingly, the cerebellum, long considered a motor structure because of its apparent function, is phylogenetically and embryologically a sensory structure. In all vertebrates it is more closely associated with sensory systems used for tracking movements of targets in the environment than it is with motor systems.