In this paper we discuss the special properties of hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one. We show that the re–expansions are independent of the inhomogeneity. We illustrate how this leads to a symmetry breaking in the Stokes constants within a pair of formal solutions of a differential equation. A consequence is that Stokes constants may exactly vanish in higher–order equations, leading to dramatic simplifications in the hyperasymptotic structures. Two examples are included.