We investigate the relationship between the notion of progressive taxation and inequality reduction under a general version of the concept of inequality equivalence. We consider a two-parameter formalization of the concept of inequality equivalence that both includes, as special cases, the intermediate inequality equivalence and the path-independent/unit-consistent inequality equivalence. Both criteria could range from relative to absolute inequality views as the parameters in the formulation change. For the path-independent/unit-consistent inequality equivalence the condition of nondecreasing average tax rate is necessary and sufficient to guarantee the inequality-reducing effect of taxation for all the inequality views in between the relative and the absolute.