Starting from the multi-local Klein-Gordon equations with Lorentz-scalar squared-mass operator we give a covariant quark representation of the general composite mesons and baryons with definite Lorentz transformation property. The phenomenologically observed hadron mass spectra is pointed out to satisfy possibly the approximate symmetry under the tilde U(4) transformation group concerning the spinor freedom of light constituent quarks, including the chiral transformation as a subgroup. This symmetry predicts the existence of new type of chiral mesons and baryons out of the conventional framework in non-relativistic quark model: For light q qbar systems, the scalar sigma- and axial-vector a1-nonets are predicted to exist as relativistic S-wave states besides the ordinary P-wave state mesons. Two "exotic" JPC=1-+ mesons are predicted to exist as relativistic P-wave states, which are possibly assigned as pi1(1400) and pi1(1600). For light quark baryons the extra 56 with positive parity and the extra 70 with negative parity of the static SU(6) are predicted to exist as the ground state chiral particles.