Here are the solutions (there may be more):
Spoiler:
I was inspired by Wilmer's 6-sided dice problem, so I created one that is slightly more complex.
4 six-sided dice have numbers on each face such that:
When the dice are rolled, every sum on the faces of the dice from 1 to 1296 is possible.
The greatest numerical value of a face, positive or negative, is 436.
Only one die has any negative values on it.
Challenge: Give one possible solution to the problem. I will post the solutions I have found later as a spoiler.