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This is the first of two papers concerning saddle-shaped solutions to the semilinear equation L K u = f (u) in R 2 m , 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 ″) ∈ R m × R m : | x ′ | = | x ″ | } , and vanish only on this set. By the odd symmetry, L K coincides with a new operator L K 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 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 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. [ABSTRACT FROM AUTHOR]