Is ὀργίζω, to anger, cognate with ὄργια, a secret rite or ritual? Wiktionary has a red link from the uncommon modern word to a not-yet-existing page for the ancient word (with accents). It seems at least plausible that these words would be linked, if the idea is that an inward state of emotion is described based on an outward cause or manifestation, as is often the case in Greek. Maybe the idea is that these rites unleashed passionate emotions whose expression was not normally socially permitted?

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It's not quite clear, but more likely ὄργια (and its rarer singular ὄργιον) are unrelated to the family of ὀργή "anger" / ὀργίζω "to anger", and are instead from the root of ἔργον "act, deed" (making them cognate with English work).

The argument is based on semantics, since the shift "act" to "ritual act, rite" is an easy one while it's harder to see a semantic path between "anger" and "ritual" (though not impossible, as you say). Here is Chantraine's etymological dictionary:

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And Beekes:

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(The reference to Chantraine 1933:55 is to a list of words in -ιον which might possibly be borrowings, but Chantraine seems no longer to have thought this possibility worth mentioning when he compiled his dictionary.)

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    Thanks, that's very helpful. LSJ says ὄργια is most frequently used to refer to the Dionysian cult, which would make for a pretty close semantic connection to the idea of releasing one's passions.
    – user3597
    Feb 27, 2021 at 23:53
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    The most likely etymology derives orgia and orgeon from the radix *werg of érdo, érgon, etc. The whole to do is about sacred acts, and one can approach this by drawing a parallel to to dromena, cf. Wilamowitz, Glaube 2, 70. Ruijgh in Minos l. c. posits *worga « acte rituel ». Not to miss the folk etymology [attributed to?] Dyonisus that the words had been associated with ὀργή.
    – vectory
    Mar 1, 2021 at 16:20

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