Abstract A likely candidate mechanism to heat the solar corona and solar wind is low-frequency “Alfvénic” turbulence sourced by magnetic fluctuations near the solar surface. Depending on its properties, such turbulence can heat different species via different mechanisms, and the comparison of theoretical predictions to observed temperatures, wind speeds, anisotropies, and their variation with heliocentric radius provides a sensitive test of this physics. Here we explore the importance of normalized cross helicity, or imbalance, for controlling solar-wind heating, since it is a key parameter of magnetized turbulence and varies systematically with wind speed and radius. Based on a hybrid-kinetic simulation in which the forcing’s imbalance decreases with time—a crude model for a plasma parcel entrained in the outflowing wind—we demonstrate how significant changes to the turbulence and heating result from the “helicity barrier” effect. Its dissolution at low imbalance causes its characteristic features—strong perpendicular ion heating with a steep “transition-range” drop in electromagnetic fluctuation spectra—to disappear, driving a larger fraction of the energy into electrons and parallel ion heat, and halting the emission of ion-scale waves. These predictions seem to agree with a diverse array of solar-wind observations, offering to explain a variety of complex correlations and features within a single theoretical framework.