The Taurus-Auriga complex is the prototypical low-mass star-forming region, and provides a unique testbed of the star formation process, which left observable imprints on the spatial, kinematic, and temporal structure of its stellar population. Taurus's rich observational history has uncovered peculiarities that suggest a complicated star-forming event, such as members at large distances from the molecular clouds and evidence of an age spread. With Gaia, an in-depth study of the Taurus census is possible, to confirm membership, identify substructure, and reconstruct its star formation history. We have compiled an expansive census of the greater Taurus region, identifying spatial subgroups and confirming that Taurus is substructured across stellar density. There are two populations of subgroups: clustered groups near the clouds and sparse groups spread throughout the region. The sparse groups comprise Taurus's distributed population, which is on average older than the population near the clouds, and hosts subpopulations up to 15Myr old. The ages of the clustered groups increase with distance, suggesting that the current star formation was triggered from behind. Still, the region is kinematically coherent, and its velocity structure reflects an initial turbulent spectrum similar to Larson's Law that has been modified by dynamical relaxation. Overall, Taurus has a complicated star formation history, with at least two epochs of star formation featuring both clustered and distributed modes. Given the correlations between age and spatial distribution, Taurus might be part of a galaxy-scale star-forming event that can only begin to be understood in the Gaia era.