Travesty came into English in the 1670s as a literary term, from French travesti, "dressed in disguise", the past participle of travestir. French took it from Italian travestire, which is Latin trans, across, plus vestire, to clothe. Etymonline quotes the old Century Dictionary on how a travesty differs from a parody: a parody changes the subject and characters and imitates the style, while a travesty keeps the characters and subject and makes the language absurd. Same people, wrong clothes. By the 1680s the verb could also be literal, to disguise someone with a change of clothes.
A candidate was white-robed, candidatus in Latin. Men running for office in Rome wore a toga that was probably whitened with powdered chalk, presumably to advertise clean intentions. The verb underneath is candere, to shine, which also gave us candid and candle.
Palliate is a cloak. Palliatus meant cloaked, from pallium, and the first English use in the 1400s was medical, easing a disease without curing it. Palliative care keeps that sense.
Hoodwink is the plainest of the lot. In the 1560s it meant to blindfold someone, and by about 1600 it meant to fool them.
Invest first meant to dress a person in the robes of an office, which survives in investiture. The money sense shows up in the 1610s in the East Indies trade, probably borrowed from Italian, and Etymonline's explanation is that you are giving your capital a new form. I prefer to picture it as putting your money in a new outfit.
Vestige is the one I would have bet on, and I would have lost. It comes from Latin vestigium, "footprint, trace", and the origin of that word is unknown. Wiktionary gives two guesses, one from a verb meaning to sweep and one from a root meaning to walk, and the Latin etymologist Michiel de Vaan says a connection with vestis, clothes, "does not make much sense". Investigate grows out of the same footprint, through vestigare, to track. So invest and investigate, which look like siblings, turn out to be a set of robes and a set of tracks.