I found an interesting site about the Phrygian cap
Interesting, indeed, wwh. History is plundered no less unconscionably for its its allusions as it is for its artifacts. Just as the Liberty Cap was appropriated by the French revolutionaries, so, it seems, the iron ring was appropriated by our engineers. To wit (from your source):
The iron ring that initiates wore is reminiscent of that iron ring
worn by Prometheus to remind him of his chains after his release by
Herakles.