Using single-crystal neutron diffraction we show that the magnetic structure Ni₃TeO₆ at fields above 8.6 T along the c axis and low temperature changes from a commensurate collinear antiferromagnetic structure with spins along c and ordering vector Q_C = (0 0 1.5) to a conical spiral with propagation vector Q_IC = (0 0 1.5 ± δ), δ ∼ 0.18, having a significant spin component in the (a, b) plane. We determine the phase diagram of this material in magnetic fields up to 10.5 T along c and show the phase transition between the low field and conical spiral phases is of first order by observing a discontinuous jump of the ordering vector. Q_IC is found to drift both as a function of magnetic field and temperature. Preliminary inelastic neutron-scattering data reveal that the spin-wave gap in zero field has minima exactly at Q_IC and a gap of about 1.1 meV consisting with a crossover around 8.6 T. Further, a simple magnetic Hamiltonian accounting in broad terms for these is presented. Our findings confirm the exclusion of the inverse Dzyaloshinskii-Moriya interaction as a cause for the giant magnetoelectric due to symmetry arguments. In its place we advocate for the symmetric exchange striction as the origin of this effect.