2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65361In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.24 pages in LaTeX, no figures. Accepted for publication in JDEDynamical Systemstext