  We develop a comprehensive theory for magnetoelectricity in magnetically
ordered quasi-2D systems whereby in thermal equilibrium an electric field can
induce a magnetization $m$ and a magnetic field can induce a polarization. This
effect requires that both space-inversion and time-reversal symmetry are
broken. Antiferromagnetic (AFM) order plays a central role in this theory. We
define a N\'eel operator $\tau$ such that a nonzero expectation value $\langle
\tau \rangle$ signals AFM order, in the same way $m$ signals ferromagnetic (FM)
order. While $m$ is even under space inversion and odd under time reversal,
$\tau$ describes a toroidal moment that is odd under both symmetries. Thus $m$
and $\langle \tau \rangle$ quantify complementary aspects of magnetic order in
solids. In quasi-2D systems FM order can be attributed to dipolar equilibrium
currents that give rise to $m$. In the same way, AFM order arises from
quadrupolar currents that generate the moment $\langle \tau \rangle$. The
electric-field-induced magnetization can then be attributed to the electric
manipulation of the quadrupolar currents. We develop a $k \cdot p$
envelope-function theory for AFM diamond structures that allows us to derive
explicit expressions for the operator $\tau$. Considering FM zincblende and AFM
diamond, we derive quantitative expressions for the magnetoelectric responses
due to electric and magnetic fields that reveal explicitly the inherent duality
of these responses required by thermodynamics. Magnetoelectricity is found to
be small in realistic calculations for quasi-2D electron systems. The
magnetoelectric response of quasi-2D hole systems turns out to be sizable,
however, with moderate electric fields being able to induce a magnetic moment
of one Bohr magneton per charge carrier. Our theory provides a broad framework
for the manipulation of magnetic order by means of external fields.
