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We prove a reflection theorem, conjectured by Nakagawa and Ohno, for thenumber of quartic rings, or pairs of ternary quadratic forms, with a givencubic resolvent. Over $\mathbb{Z}$, our results are unconditional; we alsoallow the base to be the ring of integers of a general number field,conditional on some algebraic identities that are Monte Carlo verified. We alsoestablish a reflection theorem for quartic $$-forms and $48441$-forms thatrelates them to the number of $3\times 3$ symmetric matrices with givencharacteristic polynomial. Along the way, we find elegant new results on Igusazeta functions of conics and the average value of a quadratic character over abox in a local field.  We conjecture that a reflection theorem holds for pairs of $n$-ary quadraticforms for any odd $n$, and we prove this for odd cubefree discriminant. Thisfurnishes a more satisfactory answer for a question raised by Cohen, Diaz yDiaz, and Olivier, namely whether there exist an infinite family of reflectiontheorems of Ohno-Nakagawa type.

Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms