I have a two-volume hard-bound set, published by the Air Force, containing unit histories of AAF/USAF squadrons, groups, wings, air divisions, commands and air forces. No officially- approved insigne shown for either 504th squadron or group but that doesn't really mean anything. Unit insignia - including distinctive and shoulder sleeve insignia of ground units - were very often designed by unit members. Regulations called for a design to be submitted to the Army's Institute of Heraldry for official approval. Although this was usually a mere formality - a "rubber stamp", if you will - some outfits skipped that step of the process and simply used the insignia anyway (regulations then weren't "carved in stone" as they seem to be in recent years).
One of the most popular and widely-known of the unofficial "orders" is the navy-wide "Order of the Golden Dragon"; individuals who have crossed the International Dateline on board a ship of the fleet. A "rite" or ceremony is conducted during which individuals making their very first crossing are "initiated" into the "order". The initiation procedure varies throughout the Fleet. There's a similar "order" for those who cross the Equator for the first time.