Exchange anisotropy effects in equal-spin s(1) = s(2) dimeric single-molecule magnets exhibiting C2 point- group symmetry have been studied. The Hamiltonian, which is written in two-spin 4 × 4 Pauli matrixes representation in the appropriate noncollinear local coordinates systems, has been transformed to common coordinates and used to derive energy levels and spin eigenstates. An anomalous variation of the magnetic system average spin depending on the misalignment of the local anisotropy axes has been found. Particularly, a fully nonmagnetic state of an s = 1/2 spin dimer and “silent” EPR mode has been predicted. The angular boundaries of the EPR existence have been determined for any possible mutual deflections of the local ion axes for each of the 6 possible transitions between the levels of the spin system, both in zero and in a nonzero magnetic field.