So, I've got the integral $\iiint_R$$(xy+z^2)dV$ over the set $0\le z \le 1-|x|-|y|$. So, I've gotten that this makes out a pyramid in $\mathbb{R}^3$ with corners in $(0,0,1)$, $(0,1,0)$ and $(1,0,0)$, but one thing in the solution bugs me out - It says as follows: "By symmetry, the integral of $xy$ over $R$ is $0$" and then they just simplify the equation to $f(x,y,z)=z^2$.
How can I see this right away? I'm sure it's something really elementary, but can't seem to get my head around it.