The gluten free diet has helped my 5 year old immensely. She was diagnosed with mild autism at Ohio State University's Nisonger Institute. The veteran developmental psychiatrist suggested gluten free and casein free diet. Many of these children are sensitive to the gluten, and make antibodies to the gluten. But those antibodies also ATTACK the thyroid. Lower thyroid levels= agitated, low energy, severe constipation. Severe constipation- BM every 5 days= increased permeability of colon= yeast overgrowth = decreased functional gastrointestinal flora= not absorbing proper micronutrients= sick, poor nutrition, brain malnutrition and toxin overload.
My daughter would repeat the same phrases over and over and over. She would ask my favorite color 50-100 times per day. She would repeat phrases from school 50 times within 3 hours. No spontaneous speech. She would introduce herself to a much loved Gramma. She would recite an 80 page Dr Seuss book.
Within 5-6 weeks of completely GF, spontaneous speech returned.
I allow her goat milk, as the protein and fat molecules are considerably smaller than cow's milk= easier to digest. I do not allow bananas, or cheese, as they cause her constipation. I highly recommend TACA website, too, as they are aware of the importance of GF diet. It is informative, even if you cannot follow through on all recommendations. It stands for Talk About Curing Autism.
There are many components to the cause of autism, and many components to treating it, too. And as the saying goes, if you meet one child with autism, then you have met one child with autism. What works for some may not work for others.

Regarding closing their eyes when going backwards, my daughter's need was to know what was expected, and to ensure it unfolded in an expected order. It was chaotic for her if the car or grocery cart had to back up, except to back out of the driveway. They need to know what to expect, and do not like deviation. When she was younger, I had to have both hands on steering wheel at all times, even at stop lights. Hands at 10 and 2 positions or else meltdown.
Just make sure when you deviate from their perception of normal, give them advanced notice and reason, "Uh-oh!! Mommy forgot the apples. We'd better back up the train. Hold on. Silly mommy"!

Have you had him play a game of closing his eyes? It would be interesting to see if it's the visual component or if it's more a vestibular / proprioceptive response.

Doctors don't have good evidence on it. But it can be good to try, just to see if the diet works. It does for some. Gluten-free, casein-free = GFCF.

Gluten: this comes from wheat, barley and rye. Look for a lot of gluten free products in your grocery store.

Casein: this comes from milk. Make sure you switch to soy, rice or almond milk. Look at every ingredient list on everything you buy. Make sure it contains NO milk or any byproducts that come from milk.

These diets made no difference for my kids' behavior, but the gluten-free part cleared up my daughter's psoriasis on her legs.

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That I need to go back to (fill in blank) and why. Her reasoning is better when on GFCF diet.