We investigate the existence of multiple turbulent states in highly turbulent Taylor-Couette flow in the range of Ta=10 11 to 9×10 12 by measuring the global torques and the local velocities while probing the phase space spanned by the rotation rates of the inner and outer cylinders. The multiple states are found to be very robust and are expected to persist beyond Ta=10 13 . The rotation ratio is the parameter that most strongly controls the transitions between the flow states; the transitional values only weakly depend on the Taylor number. However, complex paths in the phase space are necessary to unlock the full region of multiple states. By mapping the flow structures for various rotation ratios in a Taylor-Couette setup with an equal radius ratio but a larger aspect ratio than before, multiple states are again observed. Here they are characterized by even richer roll structure phenomena, including an antisymmetrical roll state.