Just wanted to share our family's amazing experience with cubing.
My husband has really got into the Rubik's cube in the last year or so, and started learning how to get quicker on YouTube tutorials. Then he started teaching our son (8.5, probably ASD, SPD). My son can now solve it quite quickly and completely reliably (after about three months of practising) and this has given him some much needed social cache at school as he is impressing other children.
They just entered the UK Rubik's Cube Competition and competed this past weekend. It was such a wonderful experience - such a friendly group of "competitors" - it didn't even feel like a competition, more like a friendly get together of like minded people. Lots of young men, many of whom may well be on the spectrum somewhere, all fiddling away with their cubes, all so happy to chat to young ones. It is purely meritocratic, with no age groups, so my son was in a group with a 78 year old man in HIS first competition. To my son, this is a BIG DEAL as it means not being treated like a child.
There are about 9 comps a year in various places, and lots of kids seemed to be so pleased to see each other again after a couple of months. Everyone was very VERY accepting of social awkwardness, shyness, not wanting photos taken, and so on. Everyone was totally happy to chat and share tips about how to deal with awkward cubing moments (don't ask me, I barely understand it). It was laid back, with no fanfare moments particularly.
It is also a hobby which is very stretchable, if that makes sense. Once you have mastered the 3x3 cube there are dozens of other puzzles to master and of course everyone wants to get quicker all the time. Also, there are twists like cubing one-handed, blindfolded, and so on. It is NOT MAGIC - there is a method that anyone can learn. Lots of kids age about ten and up were completely self-taught from YouTube videos - I think my son is a bit lucky having his dad teach him, but it is not the only way.
Several other mums and I shared some quite special moments watching our unique and misunderstood children just blossom in this environment. If you can go along to an open event near you just to watch sometime, it is quite eye-opening. I can't recommend it enough.