[en] The symmetric eikonal theory of electron capture by relativistic projectiles is developed. Simple analytical expressions are obtained for K-shell-K-shell charge transfer total cross sections when the spin of the electron flips as well as when it remains unchanged. Comparisons with experimental data and other theoretical relativistic approximations are given. It is also shown that the relativistic symmetric eikonal total cross sections coincide at asymptotically high impact energies with the relativistic second-order Openheimer-Brinkman-Kramers and relativistic continuum distorted wave ones. (author)