I can't tell you what this grammar rule is called because I simply don't know, but I can try and explain to you what it's actually saying. Consider the following example:
"Always help people in need" being their motto, they rushed into the burning house to save the boy.
I hope you can see that "always help people in need" being their motto structurally corresponds to death being contrary to their principles in your example. It tells you the reason why the people in this passage rushed into the burning house to save the boy. The reason was because "Always help people in need" was their motto. So, they were in a sense obliged to do what they had to do.
We can now apply the same logic to your example and ask the following question: why did they take precautions against it? The answer would be: the reason they took precautions against it was because death was contrary to their principles.
Well, I guess technically you could say that it's a type of adverbial clause that modifies the main clause. In other words, it tells you the reason why the action in the main clause is done. Honestly, that's really all I can say about this.