  In a simple PT-symmetric model we demonstrate that and how the violation of a
reflection symmetry $E_j=-E_{N+1-j}$ of the spectrum (called "self-duality" by
Dunne and Shifman) is connected with the loss of the simplicity of the shape of
the boundary of the domain D of parameters where the Hamiltonian H is
quasi-Hermitian, i.e., where all the bound-state energies are real.
