1550s, from French spiral (16c.), from Medieval Latin spiralis "winding around a fixed center, coiling" (mid-13c.), from Latin spira "a coil, fold, twist, spiral," from Greek speira "a winding, a coil, twist, wreath, anything wound or coiled," from PIE *sper-ya-, from base *sper- (2) "to turn, twist." Related: Spirally. Spiral galaxy first attested 1913.
before vowels chaet-, word-forming element meaning "hair," also, in scientific use, "spine, bristle," from Latinized form of Greek khaitē "long, loose, flowing hair" (of persons, also of horses, lions), from an old PIE word for "hair, mane," source also of Avestan gaesa- "curly hair," gaesu- "'curly haired," Modern Persian ges "hair that hangs down, curls;" Middle Irish gaiset "bristly hair."
<a href="https://www.etymonline.com/word/spirochete">Etymology of spirochete by etymonline</a>
Harper, D. (n.d.). Etymology of spirochete. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/spirochete