  The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity
relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms
(equivalently, cubic rings) of two different discriminants $D$, $-27D$; it
generalizes cubic reciprocity and the Scholz reflection theorem. In this paper,
we present a new approach to this theorem using Fourier analysis on the adelic
cohomology $H^1(\mathbb{A}_K, M)$ of a finite Galois module, modeled after the
celebrated Fourier analysis on $\mathbb{A}_K$ used in Tate's thesis. This
method reduces reflection theorems of O-N type to local identities. We
establish reflection theorems of O-N type for cubic forms and rings over
arbitrary number fields, and also for quadratic forms counting by a peculiar
invariant $a(b^2 - 4ac)$. We also find relations for the number of forms over
$\mathbb{Z}[1/N]$ and for forms of highly non-squarefree discriminant
(discriminant reduction).
  In a sequel to this paper, we will deal with reflection theorems for quartic
rings, $2\times 3\times 3$ symmetric boxes, and binary quartic forms. In these
cases the local step is much more involved.
