  As a candidate Kitaev material, Na$_2$Co$_2$TeO$_6$ exhibits intriguing
magnetism on a honeycomb lattice that is believed to be $C_3$-symmetric. Here
we report a neutron diffraction study of high quality single crystals under
$a$-axis magnetic fields. Our data support the less common notion of a magnetic
ground state that corresponds to a triple-$\mathbf{q}$ magnetic structure with
$C_3$ symmetry, rather than the multi-domain zigzag structure typically assumed
in prototype Kitaev spin liquid candidates. In particular, we find that the
field is unable to repopulate the supposed zigzag domains, where the only
alternative explanation is that the domains are strongly pinned by hitherto
unidentified structural reasons. If the triple-$\mathbf{q}$ structure is
correct then this requires reevaluation of many candidate Kitaev materials. We
also find that fields beyond about 10 Tesla suppress the long range
antiferromagnetic order, allowing new magnetic behavior to emerge different
from that expected for a spin liquid.
