We present K (2.01-2.42 μm) and H (1.5-1.8 μm) band imaging spectroscopy of the complex environment of the young stellar object, T Tau. These observations reveal at least five distinct sources of molecular hydrogen emission in the central 4×4 arcsec of the system, including knots of emission centered on the primary and infrared companion, an east-west jet, and what appears to be a Herbig-Haro object located within 3" of the binary. We detected the latter object in 11 different quadrupole lines of molecular hydrogen, as well as in two H band forbidden transitions of Fe II. The H2 spectral line ratios point to shock heating with a variety of excitation temperatures. The images also demonstrate that the Brγ emission is point-like and centered on the primary star only. We discuss the nature and origin of the line radiation, the relationship between the outflows and the stars, and the apparent lack of UV fluorescent excited H2.