[en] We propose group theory analysis to study coherent structures and pattern formation in Rayleigh-Taylor turbulent mixing. Periodicity in the plane is considered as a primary symmetry element determining the discrete group of invariance of the coherent motion. We show that the requirements of isotropy and structural stability significantly confine the types of interactions and patterns, which may occur. In a two-dimensional (2D) flow, an isotropic periodic pattern has a mirror plane of reflection, and its spatial period may increase via a binary interaction, resulting in the appearance of a super-structure with a duplicated period. In 3D flows, the isotropic periodic patterns have either hexagonal or square symmetry, the interactions are essentially multi-pole, and the growth of spatial period(s) may result in loss of isotropy. Compared to other symmetries, the hexagonal symmetry is the best to ensure isotropy and structural stability of the patterns. To maintain isotropy of the coherent motion, a balance between the growth and decay of the length scales is required