Decay pathways and photofragment anisotropy parameters have been determined for a series of rovibrational levels of the g (3) Sigma(g)(+) state in molecular hydrogen using fast-beam photofragment spectroscopy. The g (3) Sigma(g)(+)(upsilon=4) levels are found to be weakly predissociated by homogeneous interaction with the lower lying n=2 a (3) Sigma(g)(+) state and by rotational coupling with the i (3) Pi(g) state. The observed photofragment anisotropy reveals fine and hyperfine structure depolarization. The decay of a (3) Sigma(g)(+) (upsilon=5) levels is dominated by barrier tunneling. The anisotropy parameters is identify the symmetry of the continuum to be of mixed Sigma(g) and Pi(g) character, which implies that hydrogen molecules can tunnel through two different barriers. The determination of the kinetic energy and the anisotropy parameter of the photofragments have resulted in a number of spectral reassignments.