My kid likes being upside down and being spun. I always assumed it's vestibular.

sounds like a nutritional deficit or intestinal dysbiosis to me. I would definitely get an Organic Acids Test (OAT) done. It will show you exactly where the imbalances are that are driving these behaviors.

Yes, my SPD DD practices inversion a lot. She needs that pressure on top of her head - it calms her down, apparently. We're working with OT and psychologist, and it's been happening less and less at school.

My son used to do that. He stopped around age 8.

Gymnastics and karate and plenty of time on the monkey bars helps.

My child does handstands and cartwheels all the time not because she needs to be upside down but because she craves the deep pressure in her arm muscles/joints. (She has SPD and Dyspraxia)This is what her OT explained. She used to fill this need by pushing up onto her arms on the back of sofas and railings.
Our OT said my daughter could try to meet this need at school in an acceptable way by pressing her palms together in front of her body but that really isn't the same and she is embarrassed to do that in front of her classmates.

A behavior chart at school is a good idea as well as giving him a time and place to practice his new skills.

Can his teacher do a reward chart?

My DD, 5, just learned to do one and is a little obsessed. I haven't heard of her doing it in the classroom...yet!

Is he in gymnastics or yoga? Does he have an OT? Try giving him a place to do it--recess or an after school class.

Is he working with an OT? If so, I'd ask them to make a school visit. If not, I'd set up treatment.

He's 8! He has major sensory issues. But what can I do to curb it in the classroom?

No but how old is DC? Probably just a sensory thing that'll get old eventually.

This is what DS has started to do, about eighty times a day. Now his teacher is complaining that he's doing it in the classroom! Anyone's child also doing this?