We got a small home trampoline (like three foot across) and Nathan is allowed to jump quietly or sit and bounce while we have a conversation. He goes to therapy and she has TheraPutty (like silly putty but doesn't get caught in fabrics and such) - he's allowed to pull and play with the putty 100% of the time he's there, and she rarely asks for any eye contact. Nathan is 11, but he's been like this FOREVER. I think it's some version of ADHD - he is demanded to be still during school, so he's gotta get his wigglies out somehow - at home it's the trampoline most often. If it's not the trampoline, it'll be my furniture or his bed.
Taking a hard-nose line at "SIT STILL" is the worst thing for this kind of issue with Nathan. It's a bit like kids with tantrums - they can hold it together at school, but when they get home they fall apart because they know their family will still love them. Well, he holds it together all day at school and rarely disobeys directions to sit still or stay in his seat... and when he gets home he has no. more. sit. still. left. to give me.
I also saw that at six (he's 11) it wasn't as obvious what was going on. He tried harder to keep it together at home as well, and sometimes was successful. That led me to think he wasn't trying hard enough or I wasn't expecting enough. It was the opposite - I was already expecting too much.
This was our experience ... I hope it helps you a bit.