The great variety of lead halide perovskite semiconductors represents an outstanding platform for studying crystal symmetry effects on the spin-dependent properties. Access to them is granted through the optical orientation of exciton and carrier spins by circularly polarized photons. Here, the exciton spin polarization is investigated at 1.6 K cryogenic temperature in four lead halide perovskite crystals with different symmetries: (almost) cubic in FA_0.9Cs_0.1PbI_2.8Br_0.2 and FAPbBr₃, and orthorhombic in MAPbI₃ and CsPbBr₃. Giant optical orientation of 85% is found for the excitons in FA_0.9Cs_0.1PbI_2.8Br_0.2, MAPbI₃, and CsPbBr₃, while it amounts to 20% in FAPbBr₃. For all studied crystals, the optical orientation is robust to detuning of the laser photon energy from the exciton resonance, remaining constant for high energy detunings up to 0.3 eV, above which it continuously decreases to zero for detunings exceeding 1 eV. No acceleration of the spin relaxation for excitons with large kinetic energy is found in the cubic and orthorhombic crystals. This evidences the absence of the Dyakonov-Perel spin relaxation mechanism, which is based on the Rashba-Dresselhaus splitting of spin states at finite k-vectors. This indicates that the spatial inversion symmetry is maintained in perovskite crystals, independent of the cubic or orthorhombic phase.