Perhaps oral memory lasts longer when the average person lives to be 120 instead of the somewhere between 23 and 35 I would expect for the middle ages.
There is definitely some form of common mythology with mythologized but historical foundation story, that is shared among all magi.
Along with the code of hermes that is some evidence of a common hermetic culture.
I think also that the christian parts of the order of Hermes would have its own lively religious culture with different viewpoints on how christianity ought to be and what the role of magi and magic is in christianity.
There is an excellent old threat on the topic here: On the Religion of Magi
Likewise I think that the order is likely to have it own rituals of all sorts. Religious rituals, rituals of passage (the gauntlet springs to mind), all sorts of things.