A violation of time-reversal invariance of the nuclear Hamiltonian results in a violation of the reciprocity relation connecting the magnitudes of nuclear reaction cross sections in which initial and final states are interchanged. The development of reaction theory in the absence of T invariance is outlined, and the connection between T violation and reciprocity violation is calculated for the cases of direct reactions, isolated resonances, average compound-nucleus cross sections, and fluctuating cross sections measured with good energy resolution. It is found that in a direct reaction the magnitude of the reciprocity violation is proportional to the matrix elements of the T-odd part of the Hamiltonian connecting different competing residual states, divided by the energy separations of these residual states. In isolated resonances and average cross sections the effect depends entirely on the presence of a competing direct reaction. In fluctuating cross sections the rms value of the reciprocity violation is proportional to the rms absolute value of the matrix elements of the T-odd part of H connecting different "compound states," divided by the geometric mean of the average spacing and the average width of these "compound states." Thus the effect is favored in fluctuating reactions over direction reactions by a factor of the order of the ratio of the mean spacing of residual levels to the geometric mean of the average widths and spacings of compound levels. An additional strong enhancement of the fluctuating effect appears in the presence of competing strongly absorbed channels. Various aspects of possible experimental tests of reciprocity violation are discussed.