  We report our theoretical predictions on the linear magnetoelectric (ME)
effects originating from odd-parity multipoles associated with spontaneous spin
and orbital ordering on a diamond structure. We derive a two-orbital model for
$d$ electrons in $e_g$ orbitals by including the effective spin-orbit coupling
which arises from the mixing between $e_g$ and $t_{2g}$ orbitals. We show that
the model acquires a net antisymmetric spin-orbit coupling once staggered spin
and orbital orders occur spontaneously. The staggered orders are accompanied by
odd-parity multipoles: magnetic monopole, quadrupoles, and toroidal dipoles. We
classify the types of the odd-parity multipoles according to the symmetry of
the spin and orbital orders. Furthermore, by computing the ME tensor using the
linear response theory, we show that the staggered orders induce a variety of
the linear ME responses. We elaborate all possible ME responses for each
staggered order, which are useful to identify the order parameter and to detect
the odd-parity multipoles by measuring the ME effects. We also elucidate the
effect of lowering symmetry by a tetragonal distortion, which leads to richer
ME responses. The implications of our results are discussed for $5d$ transition
metal oxides, $A$OsO$_4$ ($A=$ K, Rb, and Cs), in which the order parameters
are not fully identified.
