  In 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.
