It’s apparently somewhat well established that some regions of the brain are influenced more by paternal genes (the paternal brain), and some by maternally genes (the maternal brain).
The Imprinted Brain theory of autism says that autism results from the paternal brain being more developed, and the maternal brain being less developed, with an increased paternal brain causing Aspergers syndrome, and a reduced maternal brain causing more severe autism.
The father’s genes want the mother to invest more resources in a child than the mother’s genes do. Maternal genes have more desire for child to empathize with her and siblings to make childcare less costly. Paternal genes have more desire for competition between siblings over resources.
I had previously been impressed by a theory in the book Shadow Syndromes that involves a less developed cerebellum causing a slowness to shift one’s attention as a child, which makes one less likely to notice facial expressions. The Imprinted Brain theory can imply this (the cerebellum is one of the maternal brain areas which is underdeveloped).
The evidence is hard to summarize, but here’s an example:
autism increases with paternal (and maternal) age (Gillberg, 1980), and assisted reproduction via intracytoplasmic sperm injection (ICSI) may increase the risk for syndromes of dysregulated imprinting, including Angelman and Beckwith-Weideman (Paoloni-Giacobino & Chaillet, 2004; Waterland & Jirtle, 2004; Maher, 2005). Both paternal age and ICSI are expected to contribute to methylated-gene defects, which may include effects on brain-imprinted genes (Waterland & Jirtle, 2004; Malaspina et al., 2005).
I recommend reading the discussion section of the paper, which contains much more information than I can summarize.
The paper also mentions evidence that paranoid schizophrenia is an opposite of autism (involving a highly developed maternal brain) – schizophrenics are more likely than most people to notice/imagine that someone is looking at them (see (Mentalism and mechanism and The eyes have it).
Here is an apparently unrelated argument for schizophrenia and autism being opposites.