"To date, there is no evidence that hyperbaric oxygen therapy improves core symptoms and associated symptoms of ASD [autism spectrum disorder]."
So said the results of the review by Xiong and colleagues  (open-access available here) completed under the auspices of the Cochrane Collaboration, leaders in the science and publication of systematic reviews (see here for another example).
Looking at the collected peer-reviewed science on the topic of hyperbaric oxygen therapy (HBOT) for autism - where "the patient breathes near 100% oxygen intermittently while inside a hyperbaric chamber pressurized to greater than sea level pressure", authors looked to determine several factors. Whether "treatment with hyperbaric oxygen: 1. improves core symptoms of ASD, including social communication problems and stereotypical and repetitive behaviors; 2. improves noncore symptoms of ASD, such as challenging behaviors; 3. improves comorbid states, such as depression and anxiety; and 4. causes adverse effects."
The results kinda mirrored what has already been previously described in the existing research review literature (see here), that based on one trial only  reaching their inclusion criteria, there is little evidence at the moment to say that HBOT is blanket indicated for autism.
Obviously one has to be a little careful that 'absence of science' is not construed as 'absence of evidence'. Indeed, I'm a little annoyed that the authors start suggesting that HBOT "may not be appropriate" for further study with autism in mind in light of the lack of studies in this area based on "the absence of a persuasive theory of change from experimental and clinical studies, the unknown long-term safety of the treatment, and the financial and opportunity costs of not participating in other proven therapies." No, I'm not defending HBOT as an intervention for autism but I am defending the idea that research reviews based on one study should really be trying to clarify the science rather than block future research efforts. Indeed, I'm inclined to direct you to the paper by Rossignol and colleagues  potentially answering questions such as mode of action and how one might want to look at potential best-responders to this type of intervention (autisms people, autisms). That and what the Sampanthavivat study included in the Xiong review actually said about 'safety' during their trial: "interventions were safe and well tolerated with minimal side effect from middle ear barotraumas."
To close, and in keeping with the date, a spooky song (and a rather spooky singer it has to be said...)
 Xiong T. et al. Hyperbaric oxygen therapy for people with autism spectrum disorder (ASD). Cochrane Database Syst Rev. 2016 Oct 13;10:CD010922.
 Sampanthavivat M. et al. Hyperbaric oxygen in the treatment of childhood autism: a randomised controlled trial. Diving Hyperb Med. 2012 Sep;42(3):128-33.
 Rossignol DA et al. Hyperbaric oxygen treatment in autism spectrum disorders. Medical Gas Research. 2012; 2: 16.
Xiong T, Chen H, Luo R, & Mu D (2016). Hyperbaric oxygen therapy for people with autism spectrum disorder (ASD). The Cochrane database of systematic reviews, 10 PMID: 27737490