Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function \(f\) is strongly Wright-convex of order \(n\) if and only if it is of the form \(f(x)=g(x)+p(x)+c x^{n+1}\), where \(g\) is a (continuous) \(n\)-convex function and \(p\) is a polynomial function of degree \(n\). This is a counterpart of Ng's decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.