This is the first of two papers concerning saddle-shaped solutions to the semilinear equation LKu=f(u) in R2m, where LK 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″)∈Rm×Rm:|x′|=|x″|}, and vanish only on this set. By the odd symmetry, LK coincides with a new operator LKO which acts on functions defined only on one side of the Simons cone, {|x′|>|x″|}, and that vanish on it. This operator LKO, which corresponds to reflect a function oddly and then apply LK, 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 τ↦K(τ) 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.