We consider regularizations of the convective term that preserve symmetry and conservation properties exactly. This yields a novel class of regularizations that restrain the convective production of small scales in an unconditionally stable manner Numerically, one of the most critical issues is the discrete filtering; properties required are, in general, not preserved by classical LES filters. Alternatively, here we propose to construct filters with the general form F = I + ΣMm =1 dm ˜Dm where ˜D is the discrete diffusive operator. Then, the coefficients, dm, follow from the requirement that, at the smallest grid scale kc, the damping effect to the wavevector-triple (kc, p,kc− p) interactions must be virtually independent of the p-th Fourier-mode. This allows an optimal control of the subtle balance between convection and diffusion to stop the vortex-stretching. Finally, the proposed method is tested for an air-filled differentially heated cavity of aspect ratio 4 by direct comparison with DNS reference results.