Dynamically relevant alignments are used in order to show that regions with weak vorticity are not structureless, non-Gaussian and dynamically not passive. For example, the structure of vorticity in quasi-homogeneous/isotropic turbulent flows is associated with strong alignment between vorticity ω and the eigenvectors of the rate of strain tensor λi, (especially - but not only - between ω and λ2) rather than with intense vorticity only. Consequently, much larger regions of turbulent flow than just those with intense vorticity are spatially structured. The whole flow field - even with the weakest measurable enstrophy - is strongly non-Gaussian, which among other things is manifested in strong alignment between vorticity and the vortex stretching vector Wi = ωjSij. It is shown that the quasi-two-dimensional regions corresponding to large cos(ω,λ2) are qualitatively different from purely two-dimensional ones, e.g. in that they possess essentially nonvanishing enstrophy generation, which is larger than its mean for the whole field.