We reconsider the definitions of both material symmetries and physical symmetries which aredescribed in terms of point groups, i.e. subgroups of the full orthogonal group, because these two conceptsare often confused and the classical descriptions of physical symmetry for inelastic behaviour of materials areimpracticable. All two-and three-dimensional point groups are classified into two types: compact andnon-compact. The reality of every compact point group in the description of a material or a physicalsymmetry is justified in four aspects, that is: (i) point groups characterized by a finite set of tensors, (ii)Hilbert's theorem for integrity bases, (iii) correlation between integrity bases and function bases (generalization of Wineman and Pipkin's theorem), and (iv) physical reality. The unreality of anynon-compact point group in the description of a material or a physical symmetry is proposed as a newprinciple of continuum physics. As applications, the complete sets of all classes of two-and three-dimensional point groups which describe physical symmetries for linear physical properties (such as thermalexpansion, piezoelectricity, elasticity, etc.) and for more general mechanical constitutive laws are given.