Consider a closed nonempty set \(S\) in the usual \(n\)-dimensional Euclidean space. Let \(x\) be a point in \(S\). The proximal normal cone to \(S\) at \(x\) is a geometric concept of great importance for the differentiability analysis of the boundary of \(S\). The authors establish some links between proximal normal cones to \(S\) and proximal normal cones to the closure of the complement of \(S\). As an application of the general “complementary proximal normal formula”, the authors obtain some invariance properties for a class of differential inclusions.