  Metals hosting gradually varying spatial magnetic textures are attracting
attention as a new class of inductor. Under the application of an alternating
current, the spin-transfer-torque effect induces oscillating dynamics of the
magnetic texture, which subsequently yields the spin-motive force as a back
action, resulting in an inductive voltage response. In general, a second-order
tensor representing a material's response can have an off-diagonal component.
However, it is unclear what symmetries the emergent inductance tensor has and
also which magnetic textures can exhibit a transverse inductance response. Here
we reveal both analytically and numerically that the emergent inductance tensor
should be a symmetric tensor in the so-called adiabatic limit. By considering
this symmetric tensor in terms of symmetry operations that a magnetic texture
has, we further characterize the magnetic textures in which the transverse
inductance response can appear. This finding provides a basis for exploring the
transverse response of emergent inductors, which has yet to be discovered.
