  We show that a lattice mode of arbitrary symmetry induces a well-defined
macroscopic polarization at first order in the momentum and second order in the
amplitude. We identify a symmetric flexoelectric-like contribution, which is
sensitive to both the electrical and mechanical boundary conditions, and an
antisymmetric Dzialoshinskii-Moriya-like term, which is unaffected by either.
We develop the first-principles methodology to compute the relevant coupling
tensors in an arbitrary crystal, which we illustrate with the example of the
antiferrodistortive order parameter in SrTiO$_3$.
