ABO3 perovskite-type solid solutions display a large variety of structural and physical properties, which can be tuned by chemical composition or external parameters such as temperature, pressure, strain, electric, or magnetic fields. Some solid solutions show remarkably enhanced physical properties including colossal magnetoresistance or giant piezoelectricity. It has been recognized that structural distortions, competing on the local level, are key to understanding and tuning these remarkable properties, yet, it remains a challenge to experimentally observe such local structural details. Here, from neutron pair-distribution analysis, a temperature-dependent 3D atomic-level model of the lead-free piezoelectric perovskite Na0.5Bi0.5TiO3 (NBT) is reported. The statistical analysis of this model shows how local distortions compete, how this competition develops with temperature, and, in particular, how different polar displacements of Bi3+ cations coexist as a bifurcated polarization, highlighting the interest of Bi-based materials in the search for new lead-free piezoelectrics.