However, there are at least two large studies of people with "gender dysphoria" or "Gender Identity Disorder," i.e., dudes who think they're chicks or vice versa. From Pasterski et al. (2014):
The prevalence of autistic traits consistent with a clinical diagnosis for an autism spectrum disorder (ASD) was 5.5 % (n = 3 MtF and n = 2 FtM) compared to reports of clinical diagnoses of 0.5–2.0 % in the general population.
So, autistic traits are 3 to 10 times more common among trannies than normal people. That study was for adults. Turning to children and adolescents, de Vries et al. (2010):
The incidence of ASD in this sample of children and adolescents [with Gender Identity Disorder] was 7.8% (n = 16).
Or 4 to 15 times more common among trannies, which agrees pretty well with the estimate from the study above.
Bear in mind that this only counts those individuals who meet the clinical threshold for ASD. That's just the tail of a distribution of autistic tendencies. Treating spergy-ness as a trait with a bell-shaped curve, then taking 1% of normal people but 6.5% of GID cases to have ASD, implies that the tranny average is more autistic than the normal people's average by 0.8 standard deviations. If you think of social skills as a kind of "height," then trannies are 2.4 inches "shorter" than normal folks on average. That's something you would notice in day-to-day interactions -- they would be noticeably weird at basic social skills.
I'm not sure what the causal links are, as I only read the abstracts of the studies to get the main results. I don't know what the authors propose. Neither have I thought about the etiology of trans-sexuals yet. Gays are infantilized, bi women are permanently adolescent, I'm not sure about lesbians (other than them being more mature than the other two groups), and I have no idea what aspect of development got screwed up with trannies. Now that the big autism link is there, though, maybe that'll provide a clue.