  The search for new materials for energy-efficient electronic devices has
gained unprecedented importance. Among the various classes of magnetic
materials driving this search are antiferromagnets, magnetoelectrics, and
systems with topological spin excitations. Cu$_3$TeO$_6$ is a material that
belongs to all three of these classes. Combining static electric polarization
and magnetic torque measurements with phenomenological simulations we
demonstrate that magnetic-field-induced spin reorientation needs to be taken
into account to understand the linear magnetoelectric (ME) effect in
Cu$_3$TeO$_6$. Our calculations reveal that the magnetic field pushes the
system from the nonpolar ground state to the polar magnetic structures.
However, nonpolar structures only weakly differing from the obtained polar ones
exist due to the weak effect that the field-induced breaking of some symmetries
has on the calculated structures. Among those symmetries is the $PT$
($\overline{1}'$) symmetry, preserved for Dirac points found in Cu$_3$TeO$_6$.
Our findings establish Cu$_3$TeO$_6$ as a promising playground to study the
interplay of spintronics-related phenomena.
