  In Braginskii extended magneto-hydrodynamics (ExMHD), applicable to
collisional astrophysical and high energy density plasmas, the electric field
and heat flow are described by the $\alpha$, $\beta$ and $\kappa$ transport
coefficients. We show that magnetic transport relies primarily on
$\beta_\parallel-\beta_\perp$ and $\alpha_\perp-\alpha_\parallel$, rather than
$\alpha_\perp$ and $\beta_\perp$ themselves. However, commonly used coefficient
fit functions [Epperlein and Haines, Phys. Fluids 29, 1029 (1986)] cannot
accurately calculate these quantities. This means that many ExMHD simulations
have significantly over-estimated the cross-gradient Nernst advection,
resulting in artificial magnetic dissipation and discontinuities. We repeat the
kinetic analysis to provide fits that rectify this problem. Use of these in the
Gorgon ExMHD code resolves the known discrepancies with kinetic simulations in
the literature. Recognizing the fundamental importance of
$\alpha_\perp-\alpha_\parallel$ and $\beta_\parallel-\beta_\perp$, we re-cast
the set of coefficients to find that each of them now shares the same
underlying properties. This makes explicit the symmetry of the magnetic and
thermal transport equations, as well as the symmetry of the coefficients
themselves.
