  For an asymptotically locally Euclidean (ALE) or asymptotically locally flat
(ALF) gravitational instanton $(M,g)$ with toric symmetry, we express the
signature of $(M,g)$ directly in terms of its rod structure. Applying
Hitchin$\unicode{x2013}$Thorpe-type inequalities for Ricci-flat ALE/ALF
manifolds, we formulate, as a step toward a classification of toric ALE/ALF
instantons, necessary conditions that the rod structures of such spaces must
satisfy. Finally, we apply these results to the study of rod structures with
three turning points.
