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?
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:
(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.)