The notion that autism is the opposite of schizophrenia can probably be traced to scholars Bernard Crespi and Christopher Badcock, who marshalled quite a bit of support for their incredibly elegant model in 2008:

Autistcs are often non-verbal while schizophrenics talk constantly.

Autistics tend to have a spatial IQ > verbal IQ profile, and a fluid IQ > crystalized IQ  profile, while schizophrenics tend to have a verbal IQ > spatial IQ profile and a crystalized IQ > fluid IQ profile.

Autistics tend to have good local brain processing and lots of short range brain connections, while schizophrenics tend to have good global brain processing and lots of long range connections.

Autistics tend to be hyperlexic while schizophrenics tend to be dyslexic.

Autistics tend to have hyper-glutamatergic neurotransmission with high rates of epilepsy, while schizophrenics tend to have hypo-glutamatergic neurotransmission

Autistics tend to pay less attention to gaze while schizophrenics have so much gaze sensitivity, they get paranoid everyone is looking at them

Autistics have reduced shared attention, while schizophrenics have delusions of conspiracy

Autistics tend to have reduced personal agency while schizophrenics have megalomania, delusions of reference

Autistics tend to have hypo-activation of BA8,44,46 in functional imaging studies, while these tend to be hyper-activated in schizophrenia

Autistics tend to have reduced activation of resting brain network while the schizophrenic resting brain network is increased

Autistics tend to have larger brains (at least in childhood), while schizophrenics have smaller brains

Autistics tend to preserve grey matter as they age, while schizophrenics show rapid grey matter loss.

Autistics have a larger more reactive amygdala while schizophrenics have a smaller amygdala

Autistics have relative right-hemisphere dysfunction while schizophrenics have relative left-hemisphere dysfunction

Autistics seem to have increased structural brain asymmetry while schizophrenics seem to have decreased structural functional asymmetry

Autistics suffer from literalness, inability to deceive, while schizophrenics suffer from delusional mentality, magical ideation

Autistics have reduced imagination and less symbolic play as kids while schizophrenics have complex social delusions and enhanced imagination in psychotic savants.

Autistics suffer from repetitive behavior, resistance to change, and focus on one or few aspects of the environment.  By contrast schizophrenics suffer from reduced filtering of stimulus relevance, loose associations and enhanced creativity.

Autistics tend to have relatives that major in the technical sciences while schizophrenics tend to have relatives that major in the humanities and social sciences.

If this model turns out to be scientifically correct, instead of just based on selective interpretation of the literature, then I think these two gentlemen deserve the Nobel Prize, which would be especially great since Crespi is a Canadian.

Their theory has certainly inspired a lot of the ideas on this blog, including controversial ideas Crespi and Badcock would probably be horrified by.

For more on their BRILLIANT research, I suggest you watch this great lecture by Crespi: