I'm not sure which ones are you talking about specifically. But there are some with some heavy symmettrical patterns indeed. To my eyes some are mesmerizing to watch.

The orbits are computed in real time, so yeah what you are seeing (modulo errors in my code is genuine)

There are some caveats though. Some orbits are periodic only in a rotating frame of reference.

EDIT: you can share the URL and I can see which orbits you are talking about

Like: https://www.nhatcher.com/three-body-periodic/?class=bhh_sate...