  Na$_2$Co$_2$TeO$_6$ has recently been proposed to be a Kitaev-like honeycomb
magnet. To assess how close it is to realizing Kitaev quantum spin liquids, we
have measured magnetization and specific heat on high-quality single crystals
in magnetic fields applied along high-symmetry directions. Small training
fields reveal a weak but canonical ferrimagnetic behavior below 27 K, which
cannot be explained by the zigzag antiferromagnetic order alone and suggests
coexisting N\'{e}el-type order of moments canted away from the zigzag chains.
Moderate fields in the honeycomb plane suppress the thermal transition at 27 K,
and seem to partly reverse the moment-canting when applied perpendicular to the
zigzag chains. In contrast, out-of-plane fields leave the transition largely
unaffected, but promotes another transition below 10 K, possibly also related
to canting reversal. The magnetism in Na$_2$Co$_2$TeO$_6$ is highly anisotropic
and close to tipping points between competing phases.
