"man of many turnings", is a more literal translation of polytropon than Lattimore and Fagles, and and in fact the whole line is better analysed as an original, literal translation from common dictionary glosses than a "crappy mishmash". Many people who want to give a literal gloss say "man of many turns" or "many-turned man".

> ἄνδρα μοι ἔννεπε, μοῦσα, πολύτροπον, ὃς μάλα πολλὰ πλάγχθη, ἐπεὶ Τροίης ἱερὸν πτολίεθρον ἔπερσεν·

man(acc.) me(dat.) tell(pres. imp.), Muse(voc.), many-turn(acc.), who(nom.) very(adv.) far(acc.) wandered(aor. 3sg. m.p., but note active sense of "to drive off"), after(adv.) Troy(acc.) holy(acc.) city(acc.) (he) sack(aor. 3sg.)

Glosses my own, text from https://scaife.perseus.org/reader/urn:cts:greekLit:tlg0012.t...