If I have a curve implicitly defined by say $x^2+xy+y^2=1$, then it is clear that it is symmetric about $y=x$ because if I interchange x's with y's, then I have the exact same equation.
However, how would one adapt a similar kind of mentality to show that a curve is symmetric about $y=-x$?
It seems awfully tempting to say that you replace $x$ with $-y$ and $y$ with $-x$, but I am not sure if this is valid. It's worked with the few examples I've thought of so far like $xy=1$ and $x^2+xy+y^2=1$.
Symmetry in line $y=-x$ maps point $(x,y)$ to $(-y,-x)$ (and vice versa), so you indeed can simply replace $x$ by $-y$, $y$ by $-x$ and check whether you get the same equality.