  A ferroaxial ordering, which appears without mirror symmetry parallel to an
electric axial moment, is described by a ferroic alignment of the electric
toroidal (ET) dipole rather than the conventional electric and magnetic
dipoles. Although its emergence requires neither spatial inversion nor
time-reversal symmetry breakings, unconventional transverse responses between
the conjugate physical quantities have been proposed, which are qualitatively
different from those in multiferroic systems without both spatial inversion and
time-reversal symmetries. We theoretically investigate a general relationship
between ferroaxial ordering and its characteristic response tensor. We show
that various rotational responses corresponding to an antisymmetric tensor
component are related to the ferroaxial ordering based on symmetry analysis.
Among them, we propose that second-order nonlinear magnetostriction, where the
strain is induced by a second-order magnetic field, is one of the experimental
setups to identify the ferroaxial ordering. We show its temperature and
magnetic-field-angle dependence by analyzing a fundamental $d$-orbital model
under the tetragonal symmetry.
