The microscopic spin-spin correlations in the two-dimensional (2D) layered spin-1 honeycomb lattice compound Na₂Ni₂TeO₆ have been investigated by neutron diffraction and inelastic neutron scattering (INS). In this paper, we reveal a magnetic phenomenon where the magnetic symmetry is controlled by the nonmagnetic Na layer, which is a unique feature for the studied compound Na₂Ni₂TeO₆ with respect to other Na-based layered compounds, especially A₂M₂XO₆ or A₃M₂XO₆ compounds. The honeycomb lattice of spin-1 Ni²⁺ ions, within the crystallographic ab planes, are well separated (∼5.6 Å) along the c axis by an intermediate Na layer whose crystal structure contains chiral nuclear density distributions of Na ions. The chirality of the alternating Na layers is opposite. Such alternating chirality of the Na layer dictates the magnetic periodicity along the c axis where an up-up-down-down (↑↑↓↓) spin arrangement of the in-plane zigzag antiferromagnetic (AFM) structure [characterized by the propagation vector k = (¹/₂¹/₂¹/₂)] is found. Our results, thus, provide a strong correlation between the magnetic moments in the transition metal layers and the Na-chiral order in the adjacent nonmagnetic Na layers. Additionally, the above-described commensurate (CM) zigzag AFM order state is found to coexist with an incommensurate (ICM) AFM state below the T_N ∼ 27.5K. The ICM state is found to appear at much higher temperature ∼50 K and persists down to the lowest measured temperature of 1.7 K. Our reverse Monte Carlo analysis divulges a 2D magnetic correlations (within the ab plane) of the ICM AFM state over the entire temperature range 1.7-50 K. Further, the spin Hamiltonian has been determined by carrying out INS experiments and subsequent linear spin-wave theory analysis which reveal the presence of competing in-plane exchange interactions up to the third nearest neighbors consistent with the zigzag AFM ground state, and weak interplanar interaction as well as a weak single-ion anisotropy. The values of the exchange constants yield that Na2Ni2TeO6 is situated well inside the zigzag AFM phase (spans over a wide ranges of J₂/J₁ and J₃/J₁ values) in the theoretical phase diagram. In this paper, we thus provide a detailed microscopic understanding of the magnetic correlations and divulge the intertwining magnetostructural correlations.