Just as an additional factoid. Those are not serpents wrapped around the staff but worms.
That’s a fascinating story, Alicia, but here’s what the encyclopedia
has to say about that particular symbol:
“caduceus caduceus Pronounced As: kdyooss , wing-topped staff, with
two snakes winding about it, carried by Hermes, given to him
(according to one legend) by Apollo. The symbol of two intertwined
snakes appeared early in Babylonia and is related to other serpent
symbols of fertility, wisdom, and healing, and of sun gods. This
staff of Hermes was carried by Greek heralds and ambassadors and
became a Roman symbol for truce, neutrality, and noncombatant status.
By regulation, it has since 1902 been the insignia of the medical
branch of the U.S. army. The caduceus is much used as a symbol of
commerce, postal service, and ambassadorial positions and since the
16th cent. has largely replaced the one-snake symbol of Asclepius as
a symbol of medicine.”
Another source says: "Some confusion exists between the wand carried
by Hermes and the Staff of the demigod Aesculapius which is
classically characterized as a single serpent encircling a rough hewn
tree branch. The Staff of Aesculapius is truly the more legitimate
symbol of medicine, however the Caduceus has been adopted as the more
commonly used symbol in medicine.
According to mythology, Hermes threw his magic wand at two fighting
snakes. The snakes became entwined as they stopped fighting. The
actual origin of the Caduceus is from two sources. The first was from
the Babylonia god Ningizzida and the second was from a shepherd’s
crook that was forked on top."