Correlation of sign-flipped normals

Let $X_{1}$ and $X_{2}$ be independent $N(0,1)$ random variables. Define $$\operatorname{sgn}(u)=\begin{cases}-1,&\text{if }u<0\\0,&\text{if }u=0\\1,&\text{if }u>0\end{cases}$$ Let $Y_{1}=X_{1}\operatorname{sgn}(X_{2})$ and $Y_{2}=X_{2}\operatorname{sgn}(X_{1})$. If the correlation coefficient between $Y_{1}$ and $Y_{2}$ is $\alpha$, then $\pi\alpha$ is equal to __________.