Biting forward seems to allow the whole spine to relax and fall into proper posture from lordosis, whereas biting backwards encourages the related musculature to pull the spine INTO lordosis. While individuals with recessed "mouthbreather" features are often lordotic, those with swayback posture (= the functional opposite of lordosis) usually have decent facial features. I think it's likely that lordotic individuals are mainly backward biters and swayback individuals mainly forward biters.
How do you chew? Do you think it's significant whether the mandible bites forward, upward or backward during chewing?

Interesting thoughts.
After testing a bit it seems that my bite is straight up and down with some side to side movement towards whichever side I am chewing on. I tried doing the forward and backward chewing movement but it was really uncomfortable. It doesn't seem correct or natural, at least not the way I tried it.
From my understanding the correct bite technique is something called rotary chewing in which your lower jaw swings out a little bit wider than your upper teeth at which point you bite down back towards center. The circular motion is supposedly very good for breaking down foods quickly. I don't think that forward or backward motion comes into play at all, but it could depend on your particular anatomy.
Beyond that, the only thing other that comes to mind is keeping proper head posture while chewing. I've best heard it described as being something called a 'McKenzie chin tuck' in which you stretch out the back of your neck while tucking your jaw down towards your chest without retracting it. Doing this helps me stay in a good spinal posture - that's the only connection I could make between the two as I am not sure that bite technique has any effect on the way my spine works.
Thanks for volunteering as my test subject. You're right, it may come down to individual anatomy. What about when you're standing or walking? Do you perceive a difference between holding a static clench that pushes the mandible forward vs backward?
I do perceive a difference in which way my mandible is being pushed depending on how I tilt my head. It's the same while sitting, standing or walking. I noticed that if I tilt my head upwards my mandible feels like it is being pulled backwards, and my maxilla feels like it is being pulled downwards. However, if I tilt my head downwards, my mandible feels like it is being pushed outwards, and my maxilla feels like it is being pushed up and out. The tilt of my head also has an effect on how my spine curves and where my shoulders sit which ties into your picture.
I used to always keep my head tilted upwards in order to be able to breathe easily, but ever since I started chewing and mewing, I have been able to keep a proper posture to the point where it almost feels like I am looking downwards, and am able to breathe better than ever.
Here is the article that made me aware of head posture:
http://claimingpower.com/did-we-overlook-head-posture/
From a strictly anatomical perspective, it seems like the "forward" biting posture facilitates greater overall forward movement of the maxilla. In combination with hard mewing and proper tongue posture it seems to be the ideal orientation for forward and upwards growth.