The story
From Ancient Greek. The historically "correct" form is orgastic. Nouns from Ancient Greek that end in -sm regularly form adjectives ending in -stic: for example, enthusiasm / enthusiastic, sarcasm / sarcastic. By way of counterexample, the ahistorical -mic also appears in the terms protoplasmic and cataclysmic (instead of *protoplastic and *cataclystic).
Etymology from Wiktionary, CC BY-SA 4.0.