  Nonlinear transport phenomena offer an exciting probe into the band geometry
and symmetry properties of a system. While most studies on nonlinear transport
have looked at second-order nonreciprocal responses in noncentrosymmetric
systems, the reciprocal third-order effects dominant in centrosymmetric systems
remain largely uncharted. Here, we uncover two significant contributions to
third-order charge conductivity: one affecting longitudinal resistance and
another impacting the Hall effect. We demonstrate that these previously
unexplored contributions arise in time-reversal symmetry-broken systems from
band geometric quantities such as the Berry curvature and the symplectic
connection. We prescribe a detailed symmetry dictionary to facilitate the
discovery of these fundamental transport coefficients. Additionally, we unify
our quantum kinetic results with the semiclassical wave-packet formalism to
unveil all contributions to third-order charge transport. We illustrate our
results in antiferromagnetic monolayer SrMnBi$_2$. Our comprehensive study
significantly advances the fundamental understanding of reciprocal nonlinear
responses.
