We investigate the possibility of describing the interactions of composite ("Reggeized") hadrons in space-time. We construct a formalism giving a unified description of the spin states of composite hadrons. A Lorentz family of hadron Regge trajectories allows the introduction of local hadron amplitudes. Investigation of the covariance properties of the local hadron amplitudes with respect to proper complex Lorentz transformations, space and time inversion, charge conjugation, and gauge transformations establishes the principles according to which observed families of hadron states are described in terms of representations of the complex Lorentz group. We find that covariance under these transformations necessarily demands parity doubling ("MacDowell symmetry") for meson as well as fermion trajectories. In a world of parallel Regge trajectories (unbroken Lorentz symmetry), chirality is exactly conserved. We give a manifestly Lorentz-covariant description of the propagation of composite hadrons. Explicit expressions for the composite hadron propagators are given for chiral singlet mesons and chiral doublet fermions. Examples of physical assignments are briefly discussed. It is found in particular that particles of unit baryon number transform according to representations of the complex Lorentz group with Toller quantum number $M=\frac{1}{2}$.

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