They don’t wear white hats in all forms of dress and people don’t notice the absence of something (the RAFP ‘Mars bar’) in the same way they notice the presence of something (‘CADET’).
For example, I’m a service instructor but regularly assumed to be a RAFAC SNCO: the absence of ‘RAF AIR CADETS’ under my rank (and on my TRF, along with the absence of the phallic bird) goes unnoticed.