Transport measurements in inverted InAs/GaSb quantum wells reveal a giant spin-orbit splitting of the energy bands close to the hybridization gap. The splitting results from the interplay of electron-hole mixing and spin-orbit coupling, and can exceed the hybridization gap. We experimentally investigate the band splitting as a function of top gate voltage for both electronlike and holelike states. Unlike conventional, noninverted two-dimensional electron gases, the Fermi energy in InAs/GaSb can cross a single spin-resolved band, resulting in full spin-orbit polarization. In the fully polarized regime we observe exotic transport phenomena such as quantum Hall plateaus evolving in e^{2}/h steps and a nontrivial Berry phase.