The tautological ring of M₆, ₍ is rarely Gorenstein

We prove that the tautological rings R^* (M₆, ₍) and RH^* (M₆, ₍) are not Gorenstein when g 2 and 2g+n 24, extending results of Petersen and Tommasi in genus 2. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when g=0, 1 or g 2 and 2g+n<24. The conjecture is known for g=0, 1 by work of Keel and Petersen, and we prove several new cases of this conjecture for RH^* (M₆, ₍) when g 2.