word-forming element typically representing cross as a noun, adverb (cross-examine), adjective (crossbar), and in many words a confluence of them. "There is no distinct line of division between cross as an adjective and cross as a prefix. As a prefix, it often represents the adv. cross, or the prep. cross, across." [Century Dictionary]
Old English teag, "cord, band, thong, fetter," literally "that with which anything is tied," from Proto-Germanic *taugo (source also of Old Norse taug "tie," tygill "string"), from PIE root *deuk- "to lead" (source also of Old English teon "to draw, pull, drag").
Figurative sense is recorded from 1550s. Sense of "cravat, necktie" (usually a simple one knotted in front) first recorded 1761. The railway sense of "cross-beam between and beneath rails to keep them in place" is from 1857, American English. Meaning "equality between competitors" is first found 1670s, from notion of a connecting link. Tie-breaker is recorded from 1938. The figurative old school tie (1938) in its literal sense was a necktie of a characteristic pattern worn by former students of a particular English school.
<a href="https://www.etymonline.com/word/cross-tie">Etymology of cross-tie by etymonline</a>
Harper, D. (n.d.). Etymology of cross-tie. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/cross-tie