New measurements of directed flow for charged hadrons, characterized by the Fourier coefficient v1, are presented for transverse momenta pT, and centrality intervals in Au+Au collisions recorded by the STAR experiment for the center-of-mass energy range sNN=7.7–200 GeV. The measurements underscore the importance of momentum conservation, and the characteristic dependencies on sNN, centrality and pT are consistent with the expectations of geometric fluctuations generated in the initial stages of the collision, acting in concert with a hydrodynamic-like expansion. The centrality and pT dependencies of v1even, as well as an observed similarity between its excitation function and that for v3, could serve as constraints for initial-state models. The v1even excitation function could also provide an important supplement to the flow measurements employed for precision extraction of the temperature dependence of the specific shear viscosity.