The gravitational properties of the neutrino are studied in the weak-field approximation. By imposing the Hermiticity condition, CPT, and CP invariance on the energy-momentum tensor matrix element, we show that the allowed gravitational form factors for Dirac and Majorana neutrinos are very different. In a CPT- and CP-invariant theory, the energy-momentum tensor for a Dirac neutrino of the same species is characterized by four gravitational form factors. As a result of CPT invariance, the energy-momentum tensor for a Majorana neutrino of the same species has five form factors. In a CP-invariant theory, if the initial and final Majorana neutrinos have the same (opposite) CP parity, then only tensor-(pseudotensor-)type transitions are allowed.