Why does going from 2's complement (in binary) to the positive value by completing to 1 then adding 1 work?
Suppose that your integers are $n$ bit long. One's complement changes $x$ to $2^n-1-x$, since adding $x$ to its one's complement gives $\underbrace{1\dots 1}_{n}{}_2 = 2^n-1$.

Subtracting 1 and taking one's complement changes $x$ to $2^n-1-(x-1) = 2^n-x$.

Taking one's complement and adding 1 changes $x$ to $(2^n-1-x)+1 = 2^n-x$.