Let $G_x(\cdot)$ be the distribution function of an arbitrary random variable symmetric about $0$ (zero) and $G_x^{\leftarrow}$ is the inverse function of $G_x$ then for $p \in (0, 1)$ value of $G_x^{\leftarrow}(p) + G_x^{\leftarrow}(1-p)$ is
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For any distribution symmetric about a point $\mu$, the quantile function $Q(p)$ satisfies $Q(p) + Q(1-p) = 2\mu$. In this specific case, $\mu = 0$, so the sum is $0$.