Apparently, there are two different kinds of rod with a snake/snakes coiling around it, and people easily get confused over those two.
The rod with one snake and without wings: Rod of Asclepius. It is the symbol of medicine and health care. Asclepius was extremely skilled physician from greek mythology. It was said that a snake licked his ears clean in return of some kindness and gave his the secret knowledge about medicine. Unlike the bible, snakes were sacred beings of wisdom, healing, and resurrection to the Greeks.
The rod with two snakes and with wings: Caduceus. It is the staff of Hermes, the god of travelers and trickery. The astrological symbol of Mercury ☿ is based on it. It is also used as the symbol of US Army Medical Corps. Ancient people drew an analogy between the snake that can swiftly move between underground and surface and Hermes, who can move across the boundary of mortal world and underworld and guide the spirit to afterlife. Caduceus is also believed to be capable of hypnotizing.
Remember how Matenrou members have religious and mythological theme naming?
Kannonzaka Doppo観音坂 独歩 is Guan Yin觀音菩薩, Buddhist bodhisattva associated with compassion, who watches over everything from heaven.
Izanami Hifumi伊弉冉一二三 is Izanami伊弉冉, Japanese Shinto goddess of creation and the queen of underworld.
And Jinguji Jakurai神宮寺寂雷 travels between heavenly shrine神宮 and the underworld to guide the spirit to afterlife.