  We investigate both the longitudinal thermal conductivity ($\kappa_{xx}$) and
the planar thermal Hall conductivity ($\kappa_{xy}$) in the Kitaev spin liquid
candidate of Co-based honeycomb antiferromagnet Na$_2$Co$_2$TeO$_6$ in a
magnetic field ($B$) applied along the $a$ and $a^*$ axes. A finite
$\kappa_{xy}$ is resolved for both field directions in the antiferromagnetic
(AFM) phase below the N\'eel temperature of 27 K. The temperature dependence of
$\kappa_{xy}/T$ shows the emergence of topological bosonic excitations. In
addition, the field dependence of $\kappa_{xy}$ shows sign reversals at the
critical fields in the AFM phase, suggesting the changes in the Chern number
distribution of the topological magnons. Remarkably, a finite $\kappa_{xy}$ is
observed in $B \parallel a^*$ between the first-order transition field in the
AFM phase and the saturation field, which is prohibited in a disordered state
by the two-fold rotation symmetry around the $a^*$ axis of the honeycomb
lattice, showing the presence of a magnetically ordered state that breaks the
two-fold rotation symmetry. Our results demonstrate the presence of topological
magnons in this compound in the whole field range below the saturation field.
