For your chip you would be using a "FlexPWM" block per H Bridge, The datasheet for this chip is cut down to a point where PWM channels are a single statement, So I'm afraid I will have to be light on specifics for your specific use case
Normally a single timer can drive multiple PWM channels, each of those channels will usually have both a normal output and an inverted output that you can use, this means your high side and low side can have an output,
Depending on the exact device some have allowances for fixed dead-time delays between switching state on 1 then the other. when you not using a H-Bridge driver with that functionality built in
The 3 inputs are where the driver is only high side or low side, common in 180 degree commutation control, the 6 inputs are where they are broken out separately and usually just have some safeties in the chip to prevent both high and low on at the same time (Shoot through protection), for 120 degree commutation (180 is usually a bit better)
As for how to generate your cycle, you can think of that triangle wave like the timer in your micro, counting up and down at a frequency you set up. at each timer min / max you update the values the PWM should switch at, to make this easier, you would usually set up the timer to be a division of the frequency you wish to use, in your diagram you can see it only needs to update 8/9 times per phase per cycle, normally motor controllers would have a higher PWM frequency, but it conveys the point.
Depending on how much else your juggling, and the architecture, the next step in the PWM update may be a DMA that keeps cycling around the pattern length, or may fire an interrupt to make sure it happens in a reasonable time.