Transfer sensitivity has been seen as a means of strengthening the Pigou- Dalton ``principle of transfers'', by ensuring that more weight in the inequality assessment is attached to transfers taking place lower down in the distribution. This paper examines the concept of transfer sensitivity in detail and proposes a new definition that can be usefully applied in general contexts. The definition is based on the notion of ``favourable composite transfers'' which involve a regressive transfer combined with a simultaneous progressive transfer at a lower income level. The paper proceeds to identify when one distribution can be obtained from another using a sequence of progressive transfers and favourable composite transfers, and hence when all transfer sensitive Pigou-Dalton indices agree on their pairwise inequality ranking. Since agreement occurs in some situations when Pigou-Dalton indices are not unanimous, transfer sensitivity adds power to the ``unambiguous'' inequality judgements based on the Pigou-Dalton condition and, in particular, enables distributions whose Lorenz curves intersect to be conclusively ranked.