Reported Do not swing peacocks! as an alternative, as this is what contorqueo actually means. It only has the hurling translation with words, and likely describes the arm movements associated with powerful oratory.
Actual hurling is called iacio, proicio, or traicio. You'll note the latter two have modern derivations with martial associations, namely projectile and trajectory.
This course is my first introduction to Latin. That being said, a cursory Google supports this assertion (ex. https://www.wordsense.eu/contorquere/).
If this is true... We're being taught (for example) that "Iuppiter fulmen contorquet" means "Jupiter hurls a thunderbolt", when that would be "Iuppiter fulmen iacit," while "Iuppiter fulmen contorquet" would mean something more like "Jupiter wields/brandishes a thunderbolt."
I took three years of Latin and though it's been a while I do remember that generally we used iacio/iacere/ieci/iactum (probably misspelled a verb part) for any kind of throwing. We used the 'Ecce Romani' texts and I specifically remember a line where the kid thought the innkeeper would murder him, put him in a wagon, and "stercus supra coniecit."
"Throwing at cocks", and cock-fighting, were practiced in England until the C18 (the latter may still persist, along with dog fighting, in some rural areas). French apprentices killed cats in Paris in the Middle Ages (look up The Great Cat Massacre); the mumified bodies of cats are found in old buildings as they were sometimes walled-in during construction to ward off evil.
I took Latin over 50 years ago and this is not the vocabulary you would have learned. They are obviously trying to create a living Latin that you could use on a daily basis. Now I can finally talk about hurling peacocks. I think that worrying too much about classical vocabulary may be erudite, but misses the point of what they are attempting here. They aren't teaching anyone to read the classics. The biggest error would be that probably everyday Latin wasn't like this. Just read Plautus and Terrance if you want colloquial Latin.
I wonder how many people have a use for a living Latin that they can use on a daily basis to talk about hurling peacocks and stuff. Wanting to read the classics makes more sense to me—but teaching one to do that would be beyond Duo's reach. A less ambitious goal is being able to read the inscriptions on Roman monuments, which are plentiful in much of Europe. And for this purpose the Duolingo Latin course is useless.