Congratulations for such a rare and interesting acquisition quadrans. It makes me wonder why a caduceus was used on the reverse.
Around the time of Hadrian the public baths cost about 1 quadrans for a man and 2 quadrantes for a woman. In Pompeii a surviving advertisement states the price of a cheap wine was 1 ass or 4 quadrantes. So the common people would be exchanging a lot of these coins for minor every day goods or services.
In later Antiquity the caduceus became a symbol of commerce and negotiation. The balanced form of the symbol reflects the ideal of a balanced exchange for mutual benefit.
So the intention of the symbolism on the reverse might be to remind the users of the coin that they are living in an empire where goods and services can be fairly exchanged, and who can they thank for keeping this desirable condition in place?
The portrait of their bearded emperor on the obverse is there to remind them.
I always enjoy your comment and interpretation...