Need help understanding why $Var(max(X,Y)) = Var(min(X,Y))$ if $X$ and $Y$ are iid $N(0,1)$

Question

$Var(max(X,Y)) = Var(min(X,Y))$ if $X$ and $Y$ are iid $N(0,1)$

The book solution says

• $(X,Y)$ has the same distribution as $(-X,-Y)$ since $X$ and $Y$ are iid $N(0,1)$

• it follows that $Var(max(X,Y)) = Var(min(X,Y))$

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It make sense intuitively but I'm having trouble seeing it.

1. It make sense to me that $max(X,Y) = -min(-X,-Y)$ so for example the $max(1,4) = -min(-1,-4)$

2. Then take the variance of both sides $Var(max(X,Y)) = Var(-min(-X,-Y))$

3. Then take out the negative of the RHS and we have $Var(max(X,Y)) = Var(min(-X,-Y))$

But from here I'm stuck. Thanks for your help and patience.