A framework of two-sided (TS) densities is presented for asymmetric continuous distributions consisting of two branches each with its own generating density. The framework supports the construction of distributions with positive support and a specified mode. A general expression for the Lorentz curve, depicting income inequality, is derived in terms of these generating densities. The TS beta-t family of distributions is constructed herein as an instance within that framework. Its generating densities are a half-symmetric beta distribution for its left branch and half Student-t distribution for its right branch. A novel procedure solving for its parameters given a lower quantile, an upper quantile, a modal value and a value for the conditional-value-at-risk (CVaR) with a specified confidence is derived. The procedure shall be demonstrated using publicly available US income distribution data from 2022 by ethnicity by fitting TS beta-t parameters to those data sets. The fitted distributions shall be compared to a fitted Burr XII distributions using the maximum-likelihood estimation (MLE) method. In that process a novel income-inequality metric termed dominance-index is introduced. That dominance-index compares income inequality between two income distributions, whereas the classical Gini-index evaluates income-inequality within a single income distribution.