The time-reversal symmetry is one of the main symmetries in nature. The Hamiltonian H of the system is invariant under the time-reversal operator T introduced in quantum mechanics in 1932 [1]. In particular, there is a supplementary degeneracy of energy levels of a system containing odd numbers of electrons (Kramers degeneracy) as a consequence of the timereversal symmetry. It is shown that in coordination compounds with clusters, containing odd numbers of paramagnetic ions with a half-integer spin, there exist not only the Kramers degeneracy of energy levels, but also structural distortions caused by the existence of the time-reversal symmetry. It is due to the specific structure of the corresponding four-color point groups of magnetic symmetry, which do not contain rotation axes of third and fifth orders. We have proved that instability of spin populations occurs in corners of a triangle of isotropic antiferromagnetically coupled ions with respect to weak structural distortions, when the timereversal symmetry is violated in trihomonuclear Kramers clusters. We have also proved the theorem [2], according to which the change of the sign of the exchange interaction constant in magnetic binuclear clusters is equivalent to an isomorphic substitution of one of the cluster’s ions by an ion (denoted as “time-reversed ion”), whose wave functions are time-reversed with respect to the wave functions of the unsubstituted ion. Based on this theorem, a method of synthesis of binuclear coordination compounds with predictable magnetic properties is proposed. This method consists of presenting the operator T as a product T1T2, where T1 and T2 are operators of “incomplete” time-reversal (T1HT1-1 = -H, T2HT2-1= -H), unlike T1T2, which is the operator of complete time-reversal (T1T2HT2-1T1-1 = H). The only symmetry operation, under which the Hamiltonian remain invariant, is “the operator T1 (or T2) + isomorphic substitution of one of the dimer’s ion by the time-reversed one”. In particular, if the operator H describes the intradimer exchange interaction between paramagnetic ions with electron spins S1 = S2 = 1/2, then a change of its sign is equivalent to an inversion of the singlet (S = 0) and triplet (S = 1) spin levels. The isomorphic substitution of one of the binuclear cluster ions by the time reversed ion leads to the same result. One of these transformations violates the time-reversal symmetry, while the other restores this symmetry. As a result, a transformation occurs of the antiferromagnetic exchange interaction dimer into a ferromagnetic exchange interaction one. The above theoretical predictions are confirmed by experimental data concerning binuclear coordination compounds containing Copper(II)-Copper(II) and Copper(II)-Oxovanadium(II) dimers.