I am so sorry you were hit by a car!

I do think that there are neurons called mirror neurons that reflect bilateral pain.

When I had my big toe nail removed, recently, my other foot hurt all night too! (everything has simmered down, now, after 2 weeks of upheaval). Oddly, when I had that surgery in the podiatrist's office, the next day my left HAND was terribly swollen, and continued until now. This might be due to HAE (angioedema ) which was reactive to the trauma and needles.
I found it very strange, because I didn't do anything to that hand to make it so swollen.

The RSD patients on our RSD forum talk about "spread" of their RSD (CRPS) to the opposite side of the body.
In fact there is now mirror therapy that some try to lessen this.

I also used to think that the inflammatory substances released from an injury can enter the blood stream and light up another location! Those with autoimmune tendencies, can then create antibodies to these nerve by products, and then THAT proceeds on its own to attack normal tissue elsewhere.