  Angular momentum conservation has served as a guiding principle in the
interplay between spin dynamics and mechanical rotations. However, in an
antiferromagnet with vanishing magnetization, new fundamental rules are
required to properly describe spin-mechanical phenomena. Here we show that the
N\'eel order dynamics affects the mechanical motion of a rigid body by
modifying its inertia tensor in the presence of strong magnetocrystalline
anisotropy. This effect depends on temperature when magnon excitations are
considered. Such a spin-mechanical inertia can produce measurable consequences
at small scales.
