Inequality among people involves comparisons of social indicators such as income, health, education and so on. In recent years the number of studies both theoretical and empirical which take into account not only the individual’s income but also these other attributes has significantly increased. As a consequence the development of measures capable of capturing multidimensional inequality and satisfying reasonable axioms becomes a useful and important exercise.The aim of this paper is no other than this. More precisely, we consider the unit consistency axiom proposed by B. Zheng in the unidimensional framework. This axiom demands that the inequality rankings, rather than the inequality cardinal values as the traditional scale invariance principle requires, are not altered when income is measured in different monetary units. We propose a natural generalization of this axiom in the multidimensional setting and characterize the class of aggregative multidimensional inequality measures which are unit-consistent.