Local rotational symmetry (LRS) of a particulate system is important for understanding its structure and phase transition. However, how to properly characterize LRS for this system is still a challenge as the system normally includes both ordered and disordered local structures. Herein, based on the so-called common neighbour subcluster (CNS), we proposed a method to characterize the LRS of uniform spheres packings with the packing fraction ρ ranging within 0.20 and 0.74. It was found that different fold LRSs coexist in most packings, and their maximum degree increases at ρ < 0.64, except for the 2-fold LRS held by 6-sphere CNS that continuously increases to form the fcc crystal at ρ = 0.74. The overall LRS involving all the CNSs monotonically increases with two critical changes at ρ = (0.35-0.40) and 0.64; the evolution of individual LRSs held by specific CNS groups critically changes at ρ ≈ (0.35-0.40), 0.50, 0.55-0.60, and 0.64. The physics corresponding to these critical changes has also been discussed. The findings will significantly enrich the understanding of the structural symmetry of materials including atoms and particles.