We introduce the concept of Negative Extremal Transfers (NETs), which are transfers from the poorest individuals to the richest. This family of transfers is rich enough to describe the entire space of income distributions. Our first result shows that any income distribution can be obtained as an expansion from the uniform distribution through a sequence of NETs. In other words, NETs form a mathematical basis for the space of income distributions. Our second representation theorem establishes that any inequality index can be described based on the weight it assigns to all possible NETs. When the index is additive in NETs, and their weights are separable into two parts relative to each transfer's endpoints, it is possible to isolate the contribution of each individual to the inequality measure. This allows us to observe how much importance a given inequality index attaches to poverty concerns, in addition to inequality concerns. Our NET representation theorem can serve as a guide for proposing new inequality indices (Anecdotally, we find that indices used in practice lie in a relatively small region of the index space). Practitioners will find this representation result useful for quantifying the contribution of a given quantile or subgroup to the population's inequality level and guiding policy toward the most effective transfers to lower inequality. Additionally, it enables the easy computation of index-equivalent income distributions, where subpopulations are represented by their representative earner. We also argue that the inequality index can be associated with a cardinal utility representation of each individual, which depends on her own income and the population average. This opens the possibility of tracking the evolution of subpopulation welfare over time, even when the average income is not constant.