Historical note. The adjective "cavalier" is derived from the noun "cavalier." The "caval-" part means horse, like "cavalry" (like Spanish "caballo" or French "cheval"). A cavalier is a fighter who rides a horse--a cavalryman, a knight, a mounted soldier. French, chevalier. Spanish, caballero. During the English Civil War of the 1600s, the formal names of the two sides were the "Royalists" who supported the King, and the "Parliamentarians." But they were nicknamed the Cavaliers and Roundheads. (These are proper names, so they are capitalized). The Cavaliers had a reputation for being arrogant (and/or dashing and elegant). So "cavalier" means dismissive, contemptuous, or arrogant, like the Cavaliers. The British poet Robert Browning wrote a series of poems, "Cavalier Tunes," depicting the Cavaliers as masculine, manly, and dashing: Boot, saddle, to horse, and away! Rescue my castle before the hot day Brightens to blue from its silvery grey, CHORUS.---Boot, saddle, to horse, and away!

"Twisting" here is metaphorical. It's being used to mean "to distort or corrupt." "Twisting" the English language means to distort or corrupt it. "Cavalier" = an attitude of indifference. So a "cavalier twisting" of the English language means something like, "Distoring the English language, and not caring that you're distoring it." FWIW: "cavalier twisting" here isn't a phrasal verb. It's an adverb + verb.

"cavalier" - someone who is "cavalier" about something or someone lacks respect or is very uncaring or irresponsible. So here the writer is saying the person misuses the language and does so in an extreme manner by metaphorically "twisting" the language (into knots) to try and establish that their point of view is correct. The writer is saying that people who are members of Linkedln will however understand what he is really doing.