  We prove a reflection theorem, conjectured by Nakagawa and Ohno, for the
number of quartic rings, or pairs of ternary quadratic forms, with a given
cubic resolvent. Over $\mathbb{Z}$, our results are unconditional; we also
allow the base to be the ring of integers of a general number field,
conditional on some algebraic identities that are Monte Carlo verified. We also
establish a reflection theorem for quartic $11111$-forms and $48441$-forms that
relates them to the number of $3\times 3$ symmetric matrices with given
characteristic polynomial. Along the way, we find elegant new results on Igusa
zeta functions of conics and the average value of a quadratic character over a
box in a local field.
  We conjecture that a reflection theorem holds for pairs of $n$-ary quadratic
forms for any odd $n$, and we prove this for odd cubefree discriminant. This
furnishes a more satisfactory answer for a question raised by Cohen, Diaz y
Diaz, and Olivier, namely whether there exist an infinite family of reflection
theorems of Ohno-Nakagawa type.
