  For the S-states of muonium and positronium, the hyperfine shifts to the
order $\alpha^6$ of a recently derived two-fermion equation with explicit $\cal
CPT$-invariance are checked against the results of a nonrelativistic reduction,
and the leading $\alpha^8$ shifts are calculated. An additional hyperfine
operator is discovered which can milden the singularity for $r\to 0$ of the
Dirac hyperfine operator, such that the resulting extended operator can be used
nonperturbatively. The binding correction to magnetic moments is mentioned.
