Gradient elasticity theories contain, by definition, scale parameters and therefore, naturally. that they are very attractive for modeling scale effects in the mechanics of materials with a micro-nano structure, for studying phase transformations with the formation of interphase layers that change the microstructure of materials, modified composites with nanostructures on fibers, as well as for studying connected problems of thermo-mechanics and hydrodynamics, and etc. The appearance of scale parameters in gradient models is due to the fact that not only deformations, but also their gradients are considered as arguments in the variational description of such models. As a result, the governing equations in first-order gradient models are determined not only by the tensor of elastic properties of the fourth rank, but also in the general case by elastic tensors of the fifth and sixth rank, which differ in dimension from the classical elastic moduli. The paper discusses the symmetry of the tensors of the moduli of elasticity of the sixth rank under the permutation of the indices of differentiation in the gradient elasticity, which is a consequence of the fact that the second derivatives of the displacement vector do not depend on the order of differentiation. It is noted that there are cases when, for correct formulations of applied boundary value problems, it is necessary to use in the boundary conditions the tensors of the elastic moduli of the sixth rank, symmetric when rearranging the differentiation indices (moment stresses symmetric in the last indices), even if the formally constructed versions of applied gradient theories lack this symmetry feature. It is shown that ignoring the symmetry property of the tensor of moduli of the sixth rank when rearranging the differentiation indices can lead to significant errors in comparison with correct solutions that take this feature into account.