A new presentation of the Borchers-Buchholz result of the Lorentz invariance of the energy-momentum spectrum in theories with broken Lorentz symmetry is given in terms of the properties of the Green's functions of microcausal Bose and Fermi fields. Strong constraints based on complex geometry phenomena are shown to result from the interplay of the basic principles of causality and stability in quantum field theory: if microcausality and energy positivity in all Lorentz frames are satisfied, then it is unavoidable that all stable particles of the theory be governed by Lorentz-invariant dispersion laws; in all the field sectors, discrete parts outside the continuum as well as the thresholds of the continuous parts of the energy-momentum spectrum, with possible holes inside it, are necessarily represented by mass-shell hyperboloids (or the light cone). No violation of this geometrical fact can be produced by spontaneous breaking of Lorentz symmetry, even if the field-theoretical framework is enlarged so as to include short-distance singularities wilder than distributions.