  Using the semiclassical theory of electron dynamics, we derive a
gauge-invariant expression for the spin toroidization in a periodical crystal.
We show that the spin toroidization is comprised of two contributions: one is
due to the configuration of a classical spin array, while the other comes from
the coordinate shift of the electron as spin carrier in response to the
inhomogeneous magnetic field. We then establish a direct and elengant relation
between our spin toroidization and the antisymmetric magnetoelectric
polarizability in insulators. Finally, we demonstrate our spin toroidization in
a tight-binding model and show that it is a genuine bulk quantity.
