Unusual behavior of reentrant spin-glass (RSG) compound Lu$_2$MnNiO$_6$ has been investigated by magnetometry and neutron diffraction. The system possesses a ferromagnetic (FM) ordering below 40 K and undergoes a RSG transition at 20 K. Additionally, Lu$_2$MnNiO$_6$ retains memory effect above the glassy transition till spins sustain ordering. A novel critical behavior with unusual critical exponents (β = ∼ 0.241 and γ ∼ 1.142) is observed that indicates a canting in the spin structure below the ferromagnetic transition (T$_C$). A comprehensive analysis of temperature-dependent neutron diffraction data and first-principles calculations divulge that a structural distortion induced by an octahedral tilting results in a canted spin structure below T$_C$. .