This article reports on the detailed study of the three-gluon vertex in four-dimensional SU(3) Yang-Mills theory employing lattice simulations with large physical volumes and high statistics. A meticulous scrutiny of the so-called symmetric and asymmetric kinematical configurations is performed, and it is shown that the associated form factor changes sign at a given range of momenta. The lattice results are compared to the model-independent predictions of Schwinger-Dyson equations, and a very good agreement between the two is found.