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.
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