  We present a microscopic theory of the magnetic quadrupole moment density
$\mathcal{Q}_{ij}$ in periodic crystals with combined time reversal
($\mathcal{T}$) and inversion ($\mathcal{I}$) symmetry. We obtain a
gauge-invariant expression with clear physical interpretation and demonstrate
the typical behaviour of $\mathcal{Q}_{ij}$ in a minimal two-band model that
hosts a tilted Dirac cone. We then show that $\mathcal{Q}_{ij}$ leads to an
intrinsic nonlinear anomalous thermoelectric current. As an example, we
calculate the nonlinear Nernst and Hall current in the loop-current model for
cuprate superconductors, and demonstrate their unique behaviour and capability
of indicating $\mathcal{TI}$-invariance.
