Regarding having complimentary signals:
True complimentary signals are often used for common-mode noise suppression or for other reasons, as mentioned in the comments immediately below the question.
However, the diagram provided shows slight differences in timing, with the Low side versions starting later and completing sooner than the High side. As mentioned in another answer, hysteresis or avoidance of simultaneity may be part of the reasoning for the timing difference between the High and Low signals on the same channel. Also, the drawing implies quadrature, but that may just be for the example diagram.
I am not familiar with this device, nor with what the PWM interface was designed to work. Answers to those questions may help illuminate the reason for the extra lines, and (if you are lucky) might be discussed in the processor's data sheets or app notes.
Regarding a 4-wire PWM fan, I do not believe that both lines are necessary (at least for an inexpensive computer fan).
You may this link may be useful. https://www.ekwb.com/blog/what-is-pwm-and-how-does-it-work/ It provides information about the specifics of the wires and a reasonable bit of information regarding the use of PWM in an inexpensive 4-wire computer fan.