How to show symmetry about origin of two iid pdf

Consider two independent and identically distributed discrete random variables X and Y . Assume that their common PMF, denoted by p(z), is symmetric around zero, i.e., p(z) = p(−z), ∀z. Show that the PMF of X + Y is also symmetric around zero and is largest at zero. In this question how to prove that it is symmetric about origin I have tried by taking LHS as $P_{X+Y}(\theta) = P(X+Y = \theta ) = \underset{X + Y = \theta}{\sum}P_{X,Y}(x,y)$

and RHS as

$P_{X+Y}(-\theta) = P(X+Y = -\theta ) = \underset{X + Y = -\theta}{\sum}P_{X,Y}(x,y)$ but here it is not matching can any one say how could i proceed

$$P(X+Y=-\theta)$$ $$=\sum_x P(X=x, Y=-x-\theta)$$ $$=\sum_x P(X=x) P(Y=-x-\theta)$$ $$=\sum_x P(X=-x) P(Y=x+\theta)$$ $$=\sum_x P(X=-x,Y=x+\theta)$$ $$=\sum_z P(X=z,Y=-z+\theta)$$ $$=P(X+Y=\theta)$$ where I have used the substitution $z=-x$