In approximate reasoning, it is common to modify (relax or intensify) the antecedent or the consequent of a fuzzy “If, Then” conditional. Zadeh's quantifiers based on powers of t-norms constitute the usual method to modify fuzzy propositions. Recently, a novel family of fuzzy implication functions based on powers of continuous t-norms, called T-power based implications, was introduced. This family satisfies, among other important properties, the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, it is proved that the invariance of the truth value with respect to linguistic modifiers modeled through powers of t-norms is such a strong property which allows to characterize, with some other minimal properties, the family of T-power based implications up to compositions with increasing mappings φ : [0,1] → [0,1] with φ(0) = 0.