The excitation sequence that you have shown is intended for use with a unipolar driver that uses all 6 wires of the motor.
You can hook up a 6-wire motor using only 4 wires and connect it to a bipolar driver. There are two different ways to hook it up using 4 of the six wires - either in "half-coil" or "full-coil" mode.
See Support Figures 9 and 10 for a better explanation.
From your description, it sounds like it is set up for the full-coil hookup (the winding center taps are not connected).