  The 3$d^7$ Co$^{2+}$-based insulating magnet \NCTO{} has recently been
reported to have strong Kitaev interactions on a honeycomb lattice, and is thus
being considered as a Kitaev quantum spin liquid candidate. However, due to the
existence of other types of interactions, a spontaneous long-range magnetic
order occurs. This order is suppressed by applied magnetic fields leading to a
succession of phases and ultimately saturation of the magnetic moments. The
precise phase diagram, the nature of the phases, and the possibility that one
of the field-induced phases is a Kitaev quantum spin liquid phase are still a
matter of debate. Here we measured an extensive set of physical properties to
build the complete temperature-field phase diagrams to magnetic saturation at
10 T for magnetic fields along the $a$- and $a^*$-axes, and a partial phase
diagram up to 60 T along $c$. We probe the phases using magnetization, specific
heat, magnetocaloric effect, magnetostriction, dielectric constant, and
electric polarization, which is a symmetry-sensitive probe. With these
measurements we identify all the previously incomplete phase boundaries and
find new high-field phase boundaries. We find strong magnetoelectric coupling
in the dielectric constant and moderate magnetostrictive coupling at several
phase boundaries. Furthermore, we detect the symmetry of the magnetic order
using electrical polarization measurements under magnetic fields. Based on our
analysis, the absence of electric polarization under zero or finite magnetic
field in any of the phases or after...
