The Wigner time-reversal operator T is represented as the product of two or three so-called operators of incomplete time-reversal, under the action of which not all the angular momentum projection operators change sign. It is shown that when the symmetry of time reversal is violated (reduced) in systems with Kramers degeneracy of energy levels, a violation of the Kramers theorem occurs, with the exception of one case when such reducing is insufficient to remove the Kramers degeneracy. The commutation and anticommutation relations between operators of incomplete time reversal, as well as between these operators and the operator T, are found. It is shown that these relations are different for Kramers and non-Kramers systems.