Irrespectively of breaking or non-breaking of the symmetry of a given theory, symmetric renormalization is shown to be possible without any difficulty. This is carried out by the method of mass-independent renormalization on the basis of renormalizing the effective action as a whole. The mass-independent renormalization makes it possible to renormalize simultaneously all theories with any squared mass µ2 (including µ2 < 0 for spontaneous symmetry broken theories) by the counterterms of the same form as in the arbitrary symmetric theory with µ2 = M2 > 0. The homogeneous renormalization group equation derived from this symmetric and mass-independent renormalization is shown to be equivalent to a set of generalized Callan-Symanzik equations due to the conventional renormalization method. The relations between these two renormalization methods are given generally.