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We study structural symmetries of two-dimensional (2D) photonic crystals with anisotropic unit cells, including square- and rectangular-lattices with orientationally modulated elliptic motifs, and a compound structure consisting of circles with sixfold rotational symmetry and elliptical lines with twofold symmetry, which are created through elastic deformation of a single elastomeric membrane with circular pores. We then investigate the photonic bandgap (PBG) properties of the corresponding 2D Si posts and their tolerance to the structural deviation. We find that in the compound structure the overall PBGs are dominated by the sublattice with a higher symmetry, while the total symmetry is determined by the one with a lower symmetry.