Anisotropic flow coefficients, v_n, non-linear flow mode coefficients, χ_n,mk, and correlations among different symmetry planes, ρ_n,mk are measured in Pb-Pb collisions at sNN = 5.02 TeV. Results obtained with multi-particle correlations are reported for the transverse momentum interval 0.2 < p_T< 5.0 GeV/c within the pseudorapidity interval 0.4 < |η| < 0.8 as a function of collision centrality. The v_n coefficients and χ_n,mk and ρ_n,mk are presented up to the ninth and seventh harmonic order, respectively. Calculations suggest that the correlations measured in different symmetry planes and the non-linear flow mode coefficients are dependent on the shear and bulk viscosity to entropy ratios of the medium created in heavy-ion collisions. The comparison between these measurements and those at lower energies and calculations from hydrodynamic models places strong constraints on the initial conditions and transport properties of the system. [Figure not available: see fulltext.].