  We demonstrate that two phenomena, the kinetic magneto-electric effect and
the non-linear Hall effect, are universal to polar metals, as a consequence of
their coexisting and contraindicated polarization and metallicity. We show that
measurement of the effects provides a complete characterization of the nature
of the polar metal, in that the non-zero response components indicate the
direction of the polar axis, and the coefficients change sign on polarization
reversal and become zero in the non-polar phase. We illustrate our findings for
the case of electron-doped PbTiO$_3$ using a combination of density functional
theory and model Hamiltonian-based calculations. Our model Hamiltonian analysis
provides crucial insight into the microscopic origin of the effects, showing
that they originate from inversion-symmetry-breaking-induced inter-orbital
hoppings, which cause an asymmetric charge density quantified by odd-parity
charge multipoles. Our work both heightens the relevance of the kinetic
magneto-electric and non-linear Hall effects, and broadens the platform for
investigating and detecting odd-parity charge multipoles in metals.
