  This is the first of two papers concerning saddle-shaped solutions to the
semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a
linear elliptic integro-differential operator and $f$ is of Allen-Cahn type.
  Saddle-shaped solutions are doubly radial, odd with respect to the Simons
cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$,
and vanish only on this set. By the odd symmetry, $L_K$ coincides with a new
operator $L_K^{\mathcal{O}}$ which acts on functions defined only on one side
of the Simons cone, $\{|x'|>|x''|\}$, and that vanish on it. This operator
$L_K^{\mathcal{O}}$, which corresponds to reflect a function oddly and then
apply $L_K$, has a kernel on $\{|x'|>|x''|\}$ which is different from $K$.
  In this first paper, we characterize the kernels $K$ for which the new kernel
is positive and therefore one can develop a theory on the saddle-shaped
solution. The necessary and sufficient condition for this turns out to be that
$K$ is radially symmetric and $\tau\mapsto K(\sqrt \tau)$ is a strictly convex
function.
  Assuming this, we prove an energy estimate for doubly radial odd minimizers
and the existence of saddle-shaped solution. In a subsequent article, part II,
further qualitative properties of saddle-shaped solutions will be established,
such as their asymptotic behavior, a maximum principle for the linearized
operator, and their uniqueness.
