  Several optical experiments have shown that in magnetic materials the
principal axes of response tensors can rotate in a magnetic field. Here we
offer a microscopic explanation of this effect, and propose a closely related
DC transport phenomenon -- an off-diagonal \emph{symmetric} conductivity linear
in a magnetic field, which we refer to as linear magneto-conductivity (LMC).
Although LMC has the same functional dependence on magnetic field as the Hall
effect, its origin is fundamentally different: LMC requires time-reversal
symmetry to be broken even before a magnetic field is applied, and is therefore
a sensitive probe of magnetism. We demonstrate LMC in three different ways: via
a tight-binding toy model, density functional theory calculations on MnPSe$_3$,
and a semiclassical calculation. The third approach additionally identifies two
distinct mechanisms yielding LMC: momentum-dependent band magnetization and
Berry curvature. Finally, we propose an experimental geometry suitable for
detecting LMC, and demonstrate its applicability using Landauer-B\"{u}ttiker
simulations. Our results emphasize the importance of measuring the full
conductivity tensor in magnetic materials, and introduce LMC as a new transport
probe of symmetry.
