Generally speaking a classic behavior of the ASD individual is to seek some sort of repetitive stimulation. There is an assumption that this feeds the reward system in their brain (although I have not seen substantiative research to suggest it). Often times this stim (as it is referred to) involves a proprioceptive motions, that is stimulation at a joint or muscle. The characteristic hand flap or sometimes a winding arms, etc. But because this is a spectral disorder, it can also border on trying to get cutaneous stimulation, like head banging. I have never heard of anyone characterizing this later one as associated with a reward system. Some of these can almost be seen as involuntary tics similar to those seen in Tourette Syndrome. Although the verbal tic seen in Tourette is a little different in ASD individuals. In this latter group, the verbal "tic" is often referred to as echolalia, where the person repeats a phrase or a word over and over and over. Something he or she heard - reward behavior.
This can be particularly tough on an ASD individual because it is thought to be associated with reward, it is hard to automatically suggest that such behavior should be extinguished through therapy. In the early days, we were told to work on those behaviors and I believe the schools tried to do the same. But, imagine you could not get any reward for a job well done (school work). In time your frustration may build up. And, since most of them do not have coping skills to deal with this frustration it is hypothesized that some outburst and meltdowns may be associated with trying to extinguish these stims and the like.
Of course, it also shines a bright light upon the oddity of these individuals. My son will at times skip throughout the house repeating word for word some video game review he heard on Youtube. And, even if the review was 30 minutes long he could repeat it word for word after hearing it only once. It would not make any difference if only family were home or someone came over to visit. He would conduct this activity without any sense of understanding social norms. This would happen in place of stopping for the normal greeting.
Some Autistic kids have muscle weakness along with this strong desire to stim their proprioceptive sensory system.
On thing we have found that really helps reward our son while trying to extinguish these odd behaviors, have him get on a regular "diet" of lifting weights. This puts a lot of stimulation on their joints while also building muscle. We have only been at this less than a month, and already we have seen some improvements in reducing some of his "socially unacceptable" stim patterns and even helping to reduce some of his outburst. It has not eliminated everything and we are sober enough to understand it may never, but it seems to be a good fit for him at the moment.
It fits well with our lifestyle of trying remain active ourselves (fitness wise).