But do they?
Giacomo Rizzollati and colleagues (1992) conducted experiments on monkeys, recording signals from parts of the frontal lobes which are concerned with motor commands. The researcher found that there are cells that fire not only when the monkey performs certain specific movements but also when the monkey watches another monkey performing the same action.
‘Monkey-see monkey-do neurons’
Rizzollati terms these neurons mirror neurons (‘monkey-see monkey-do neurons’). Thanks to the famous neuroscientist V.S. Ramachandran (1995) who found the mirror neurons theory very compelling and ground-breaking, it became widely known and inspired many researchers to explore it further.
Ramachandran states that that “mirror neurons would do for psychology what DNA did for biology: they will provide a unifying framework and help explain a host of mental abilities that have hitherto remained mysterious and inaccessible to experiments… With knowledge of these neurons, you have the basis for understanding a host of very enigmatic aspects of the human mind: "mind reading" empathy, imitation learning.”
‘Broken mirror neurons’ hypothesis and autism
Ramachandran suggests that “a loss of the mirror neurons may explain autism… Without these neurons, the child can no longer understand or empathize  with other people emotionally and therefore completely withdraws from the world socially.”
It has been suggested that the mirror neurons are dysfunctional in autism (Dapretto et al. 2006; Hadjikanti et al. 2007; Lee et al 2006). Professor Tantam (2009) remains sceptical about the role of mirror neurons in ASD and cites studies to show that the evidence is inconclusive: there are research findings for an impaired ability to imitate (Smith & Bryson 2007) but also for an unimpared ability (Bird et al. 2007) and some evidence that mirror neurons are intact in autism (Hamilton, Brindley, Frith 2007) .
Thus, while the broken mirror hypothesis predicts that autistic individuals should show severe impairments in understanding and imitating actions, many research studies have found no such impairments. In fact, autistic individuals may be very good imitators. For some, it is vital to see other people doing certain activities in order to be able to perform these activities, for example:
As usual I used my companion (in this case my mother) as if she were a reflection of myself. This assured me that what I understood my body to be doing was exactly what it was doing, just as I liked to be sure that I was an actual part of the person whom I was using to model my own existence, by insisting that the other person and I performed similar actions. This occasionally appeared in every day life by my either eating or drinking to catch up exactly with the other person or, more inconveniently, by insisting that he or she emptied her glass or cleared his plate to exactly the same point that I had reached if I were ahead. It was only by seeing what I had done actually happen that I could be sure that I was doing (Lucy Blackman).
Besides, many people with an ASD are excellent mimics – able to take another person’s way of speaking, moving, etc. (Tantam 2009) in order to disguise their difficulties in understanding social and communicative conventions, e.g.:
I had been… able to mimic sound or movement without any thought whatsoever about what was heard or seen. Like someone sleep-walking and sleep-talking, I imitated the sounds and movements of others – an involuntary compulsive impressionist. This meant that I could go forward as a patchwork façade condemned to live life as ‘the world’ caricature. (Donna Williams)
[When all else failed] I used to rely on a ‘fitting in’ trick that is nothing more than a sophisticated form of echolalia. Like a professional mimic I could catch someone else’s personality as easily as other people catch a cold. I did this by surveying the group of people I was with, then consciously identifying the person I was most taken in by. (Liane Holliday Willey)
Sometimes I feel like my interactions with the environment involve many little pieces of other personalities I come in contact with. These little bits get compressed and employed here and there (Stephen Shore).
PS: Body awareness problems
For some autistic individuals (especially, those with body awareness difficulties), imitation is more difficult. For example, when he was small, Naoki Higashida used to wave goodbye with his palm facing inwards, until one day he saw himself in a mirror – that’s when he realised that he was waving goodbye to himself!
 In fact, empathy in autism is experienced and expressed differently.
 See a systemic review that demonstrates that despite over a decade of research, the broken mirror theory of autism cannot be supported… Interventions based on this theory are unlikely to be helpful.