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As is well known, the single-site anisotropy vanishes in the spin-12 compounds as a consequence of the fundamental Kramers degeneracy. We argue, rather generally, that a similar property holds for the magnetically induced electric polarization P, which should depend only on the relative orientation of spins in the bonds but not on the direction of each individual spin. Thus, for insulating multiferroic compounds, P can be decomposed in terms of pairwise isotropic, antisymmetric, and anisotropic symmetric contributions, which can be rigorously derived in the framework of the superexchange (SE) theory, in analogy with the spin Hamiltonian. The SE theory also allows us to identify the microscopic mechanism that stands behind each contribution. The most controversial and intriguing one-concerning the form, appearances, and implications for the properties of real compounds-is the antisymmetric or spin-current mechanism. In this work, we propose that, within the SE theory, the disputed magnetoelectric (ME) properties of tetragonal Ba2CuGe2O7, representing the lattice of magnetic Cu2+ ions in the tetrahedral environment, can be explained solely by the spin-current mechanism, while other contributions are either small or forbidden by symmetry. First, after analysis of the symmetry properties of the SE Hamiltonian and corresponding parameters of electric polarization, we explicitly show how the cycloidal spin order induces the experimentally observed electric polarization in the direction perpendicular to the tetragonal plane, which can be naturally explained by the spin-current mechanism operating in the out-of-plane bonds. Then, we unveil the previously overlooked ME effect, where the application of the magnetic field perpendicular to the plane not only causes the incommensurate-commensurate transition, but also flips the electric polarization into the plane due to the spin-current mechanism operating in the neighboring bonds within this plane. In both cases, the magnitude and direction of P can be controlled by rotating the spin pattern in the tetragonal plane. Our analysis is based on a realistic spin model, which was rigorously derived from first-principles electronic structure calculations and supplemented with a new algorithm for the construction of localized Wannier functions obeying the crystallographic symmetry of Ba2CuGe2O7. © 2020 American Physical Society.