Let $X \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number such that

(i)	\begin{equation} Y : = - X \end{equation}

The first claim is established in R3923: Standard gaussian random real number is symmetric about zero. For the second claim, we have \begin{equation} \mathbb{P}(X X \geq 0) = \mathbb{P}(X^2 \geq 0) = 1 \end{equation} and \begin{equation} \mathbb{P}(Y X \geq 0) = \mathbb{P}(- X^2 \geq 0) = 0 \end{equation} $\square$