Aims. The immediate vicinity of T Tauri was observed with the new high-contrast imaging instrument SPHERE at the VLT to resolve remaining mysteries of the system, such as the putative small edge-on disk around T Tauri Sa and the assignment of the complex outflow patterns to the individual stars. Methods. We used SPHERE IRDIS narrow-band classical imaging in Paβ, Brγ, and the ν = 1−0 S(1) line of H 2 , as well as in the nearby continua to obtain high spatial resolution and high-contrast images over the near-infrared spectral range. Line maps were created by subtracting the nearby continuum. We also reanalyzed coronagraphic data taken with SPHERE's integral field spectrograph in J-and H-band with the goal of obtaining a precise extinction estimate to T Tauri Sb, and of verifying the recently reported claim of another stellar or substellar object in the system. Results. A previously unknown coiling structure is observed southwest of the stars in reflected light, which points to the vicinity of T Tauri N. We map the circumbinary emission from T Tauri S in J-and H-band scattered light for the first time, showing a morphology that differs significantly from that observed in K-band. Molecular Hydrogen emission is found southwest of the stars, near the coiling structure. We also detect the H 2 emitting region T Tauri NW. The motion of T Tauri NW with respect to T Tauri N and S between previous images and our 2014 data provides strong evidence that the southeast-northwest outflow triggering T Tauri NW is likely to be associated with T Tauri S. We also present accurate relative photometry of the stars, which confirms that T Tauri Sa is brightening again. Our analysis rules out the presence of the recently proposed companion to T Tauri N with high confidence.