One recent study (Falter et al., 2008) compared groups of autistic and typically developing schoolchildren on three visual/spatial tasks that normally show a male advantage: mentally rotating three-dimensional objects, positioning a cursor over a moving image, and spotting a shape hidden somewhere in a larger pattern or line drawing, or "figure disembedding". They also measured the children's second-to-fourth digit ratios, as an indicator of prenatal testosterone exposure.
Their results were not consistent with what they predicted based on the extreme-male-brain theory of autism: the autistic children did better at figure disembedding, worse at targeting the moving image, and were slightly faster (but not more accurate) at mental rotation. On further analysis, though, they seemed to be faster at different parts of the (complex, multi-step) mental-rotation process than non-autistic men seem to be. The researchers were able to break down the results of the mental-rotation task into "rotational" and "non-rotational" components --- that is, to separate the actual visualization of the object rotating from the other factors, like comparing the rotated object in your head to the one shown on the screen and deciding whether they are the same. Autistic people seem to be faster at the non-rotational parts of this process, while non-autistic, male people seem to be faster at the rotational part. So, different cognitive skills underlie the two groups' respective advantages at the same complex task.
Finally, testosterone seemed to play no role whatsoever in predicting a child's success at either mental rotation or figure disembedding; the only task that showed a relationship between digit ratios and performance on that task was targeting, which 1) was harder for the supposedly "hyper-masculine" autistic boys, and 2) was easiest for the boys with middling digit ratios --- the ones with very low, "masculinized" digit ratios performed worse, as did the ones with high, "feminized" digit ratios. There were also no group differences in digit ratio between autistic and non-autistic boys.
The term "visuospatial abilities" covers a lot of ground, and might include different skill sets depending on where and how it's being used.
For example, one of the most-studied (and most reliably replicated) cognitive differences between the sexes is a male advantage in the ability to rotate three-dimensional images on one's head; other spatial skills that tend to show a male advantage are aiming, predicting a projectile's trajectory . If autism is the same thing as having an "extreme male brain," you would expect autistic people to do better at these tasks than non-autistic people of their same gender.
There is also a well-established constellation of visual and spatial skills that autistic people tend to be better at than non-autistic people: the Block Design subscale of the Weschler intelligence tests, spotting a shape hidden inside a larger pattern (i.e., disembedding a figure), distinguishing a target figure from a crowd of "distractor figures" that are similar but not identical to the target, and reproducing a distorted or impossible image. It would be equally instructive, as far as investigating the extreme-male-brain hypothesis is concerned, to look for sex differences in how non-autistic people do on these tasks; you would also want to look for a relationship between those abilities and prenatal testosterone exposure. If the autistic cognitive style is essentially the same as the masculine one, you would predict exposure to higher levels of testosterone in utero would correlate with higher scores on the Block Design, faster and more accurate performance on visual-search tasks (like the Embedded Figures test or the tasks described in this study), and greater accuracy on drawing tasks, like the perspective-drawing task described here or the shape-drawing task described here (PDF).
They found no significant differences between the autistic children and the typically-developing children in terms of digit ratio; depending on how the study participants were grouped (because each skill was tested in a differently-sized subset of the total study population), either the autistic half or the neurotypical half of a given subgroup might have slightly higher 2D:4D. These differences were so slight as to be statistically insignificant, regardless.
Overall, the autistic children did better at both mental rotation of 3-D objects (measured by having the children turn a computer-generated image of a solid to match the view of that solid the computer showed them) and figure disembedding than the typically-developing children; they were both faster and more accurate at these tasks. (Although, for the mental-rotation task, the difference was very slight). However, they performed worse than the typically-developing children at the targeting task, in which they had to position a cursor over an image that would appear at random points within certain regions of the computer screen*. Also, the mental-rotation results are a bit more complicated than just looking at the mean reaction times and accuracies of both groups would lead you to believe. The study authors did a linear regression of each of those two outcome variables with the degree of rotation (i.e., how far from its original position did the participants have to rotate the object?), which apparently allowed them to separate out the different cognitive processes used in mental rotation:
Only the linear regression for reaction time showed any difference between the autistic and typically-developing groups, and there the difference was only in the intercept; the slopes of the two groups' lines were more or less the same.
Across participants, the degree of rotation showed a strong linear relationship with reaction time, RT (R = .99, F(1,3) = 227.88, p = .001), and accuracy scores, ACC
(R = .98, F(1,3) = 59.84, p = .004). Accordingly, consistent with previous studies of mental rotation, RT and ACC were each regressed linearly against angle of rotation for each participant, to yield a rotation slope and an intercept. The slope for each participant indexed the speed with which they mentally "rotated" objects in degrees per second. The intercept yielded by these regressions indexed non-rotational aspects of performance, presumably related to the speed with which participants mentally compare three-dimensional objects as well as decision making and response variables.
(Figure 1, in Falter et al., 2008 --- graph showing the linear relationship of degree of rotation with reaction time)
The visuospatial tasks included in this study were fairly similar to the ones Falter, Plaisted and Davis used: there was a figure-disembedding task (the Preschool Embedded Figures Test, in which a triangle is hidden somewhere in various line drawings of familiar objects), a block-building task, in which the person giving the test builds structures of varying complexity out of 1-inch cubes, and the child is supposed to build the same thing. It's not the same as the Block Design test, but it seems pretty close to it to me. The one thing Falter et al. test that Finegan et al. do not is targeting, and Finegan et al. test a few things Falter et al. don't: picture-puzzle solving (which I guess is also analogous to Block Design) and geometric-form copying (analogous to the drawing tasks I mentioned above, in the paragraph about autistic people's visuospatial strengths).
That study found no significant sex differences in any of the skills being investigated; it also found no relationship between prenatal testosterone and performance of three of the four tasks I describe above --- figure disembedding, puzzle solving and shape drawing were all independent of prenatal testosterone exposure in both sexes. For girls, there was a significant negative relationship between prenatal testosterone and scores on the block-building task --- the higher female scorers on this test tended to have lower levels of prenatal testosterone exposure than their lower-scoring peers; for boys, there was a small trend in the opposite direction --- with high scorers tending to have higher prenatal testosterone levels --- but that relationship wasn't statistically significant.
Falter et al.'s results tell me that the area of overlap between 1) the things men are usually found to better at than women, and 2) the things that autistic people are usually found to be better at than non-autistic people, within the larger domain of visuo-spatial skills, is fairly small. Also, the two "masculine" skills that autistic people tend to do better at --- figure disembedding and mental rotation --- are not at all correlated with an indirect measure of prenatal androgen exposure.
Because there is some uncertainty around the use of digit ratios as a marker for prenatal testosterone exposure, I wanted to find a study of similar cognitive abilities that measured that variable directly. Finegan et al.'s study was the only one I could find that did this, even though it was old and didn't have any autistic participants. Their results showed that prenatal testosterone has no effect on young children's performance at most visuospatial tasks, and, on the tasks it does influence, it seems to affect boys and girls differently, with higher levels of testosterone exposure boosting boys' scores but lowering girls'.
*It doesn't surprise me at all that autistic people would be slower at that, given our difficulties with motor planning. Even apart from that, some studies have found that we are not as good as neurotypical people in tracking moving visual stimuli --- see this review (full text here) for more detail.
**I use "boys" here because there were a total of two girls --- one in the autistic group, and the other in the control group --- in this study population of fifty-nine children.
Falter CM, Plaisted KC, & Davis G (2008). Visuo-spatial processing in autism--testing the predictions of extreme male brain theory. Journal of autism and developmental disorders, 38 (3), 507-515 PMID: 17674175
Finegan, J., Niccols, G., & Sitarenios, G. (1992). Relations between prenatal testosterone levels and cognitive abilities at 4 years. Developmental Psychology, 28 (6), 1075-1089 DOI: 10.1037/0012-16184.108.40.2065
Voyer D, Voyer S, & Bryden MP (1995). Magnitude of sex differences in spatial abilities: a meta-analysis and consideration of critical variables. Psychological bulletin, 117 (2), 250-270 PMID: 7724690