What Greek symbols were used to create familiar medical emblems?

A week ago, when I visited my daughter in Orange, she showed me a golden ring with two snakes intertwining with each other. I asked, "why snake?" Then I remember the snake twirling around a staff as a symbol of Medicine. My late father had this badge on his military uniform being an Army Medic Man. That's how he was when he served in the Korean War. When I was little, he was our own Family Doctor, so to speak. He also acted as our Family Dentist. He took good care of our teeth: cleaning and extracting before one crown get decayed. At the military camp in Cagayan de Oro where he worked, and where my siblings and I occasionally raided him for a treat at the PX without our poor mother's knowing, we would also find him busy compounding and preparing medicine in their Pharmacy Laboratory. So he was definitely a Jack of all Trades. Later in life, it was through our father's prodding that I should study Pharmacy - kind of traditional hand-over of skills or knowledge which most Filipinos are known about. The truth is, my parents' dreams and hopes where all centred to our eldest brother. Our father especially wanted him to become a doctor. But that dream was quenched for lack of interest on our brother's part. Their world fell apart and so was ours (the siblings)! Culture-wise, the professional parents oftentimes would goad, influence and persuade their children to follow their footsteps as a way of passing their baton of pride and prestige as in Kingship or Monarchy through their heirs to the throne. Back to the ring, what I was trying to tell my daughter in regards to the snake of her ring was - the snake is used in Medicine as its icon or symbol as mortar and pestle is used as a symbol of Pharmacy. Then we both tried to recall the term of that snake symbol which none of us could remember until I was able to find this piece of information from my online subscription of the Inspiration Line's Fascinating Facts from Around the World!

The serpentine staff used as the medical emblem is called a caduceus. It has Greek origins — Hermes, the messenger of the gods, carried it as a symbol of peace. Ancient Greeks created the caduceus as a badge of honor; ambassadors and noblemen carried a long staff entwined with garlands or ribbons to announce their presence. The garlands were later interpreted as snakes, and a pair of wings was added to denote Hermes, the winged messenger. Here's where it gets tricky. The U.S. Army medical corps adopted the caduceus as their insignia because of its similarity to the staff carried by Asclepius, the god of medicine. The staff of Asclepius is considered the "true symbol of medicine" — it features only one snake, and no wings. It's the emblem of the American Medical Association. The two symbols are quite similar in appearance, and both are derived from Greek mythology.