jmc

The sum of all integers $y$ for which $-26\le y\le 26$ is 0 since for every negative term there is a positive term of the same absolute value. Thus, the sum of all integers $x$ for which $-30\le x\le 26$ is $-30-29-28-27=-30\times 4+1+2+3=-120+6=\boxed{-114}$.