  A semiclassical theory for the orbital magnetization due to adiabatic
evolutions of Bloch electronic states is proposed. It renders a unified theory
for the periodic-evolution pumped orbital magnetization and the orbital
magnetoelectric response in insulators by revealing that these two phenomena
are the only instances where the induced magnetization is gauge invariant. This
theory also accounts for the electric-field induced intrinsic orbital
magnetization in two-dimensional metals and Chern insulators. We illustrate the
orbital magnetization pumped by microscopic local rotations of atoms, which
correspond to phonon modes with angular momentum, in toy models based on
honeycomb lattice, and the results are comparable to the pumped spin
magnetization via strong Rashba spin orbit coupling. We also show the vital
role of the orbital magnetoelectricity in validating the Mott relation between
the intrinsic nonlinear anomalous Hall and Ettingshausen effects.
