The sum comes out zero because it is, in fact, zero. Each term in the desired sum is
Each of the indices is +-1. Their sum is either even or odd.
When it's even, the term contributes (-1)^even = +constant to the desired sum.
When it's odd, the term is contributes (-1)^odd = -constant to the desired sum.
There are 2^6 instances of (iw+jw+k+l+p+q), and half of those sums are even, half odd as you can check. So the desired sum ends up as zero.
It would be better if you went with Walter's recommendation and used something other than 'sum' for the sum variable. 'Sum' would work fine.