  An eight-component formalism is proposed for the relativistic two-fermion
problem. In QED, it extends the applicability of the Dirac equation with
hyperfine interaction to the positronium case. The use of exact relativistic
two-body kinematics entails a CP-invariant spectrum which is symmetric in the
total cms energy. It allows the extension of recent \alpha^6 recoil corrections
to the positronium case, and implies new recoil corrections to the fine and
hyperfine structures and to the Bethe logarithm.
