For years, both the AXOs and Tri-Delts insisted that their Badge was on the moon. Mr. Armstrong's first wife is an AXO (his second wife is the Tri-Delta), and from what I've read, he was still married to her during the time of the moon landing. If there was a Badge on the moon, I'd say that it would be the AXO one.
But --
it's just an urban legend! AXO Headquarters has already confirmed this, and I'm thinking those at Tri-Delta would say the same thing.