Main Question
Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be a function which satisfies the property:
\begin{equation} y\ge F_1(x',y')x+F_2(x',y')\Leftrightarrow y'\ge F_1(x,y)x'+F_2(x,y) \end{equation}
where $x,y,x',y'\in\mathbb{R}$ are arbitrary and $F_1$ and $F_2$ denote the two components of $F$. Does it follow that $F(x,y)=(x,-y)$?
Motivation
For $(a,b)\in\mathbb{R}^2$, define $H_{a,b}=\{(x,y): y\ge ax+b\}$, and define a positive half space to be any set of this form. There is a correspondence $T$ between points $(x,y)\in\mathbb{R}^2$ on the one hand and positive half-spaces on the other, given by $T(x,y)= H_{x,-y}$. This correspondence preserves inclusions in the sense that
$$(x,y)\in T(x',y')\Leftrightarrow (x',y')\in T(x,y)$$
This is an instance of basic point-line duality, except applied to half-spaces instead of lines, so it may be better to call it "point-halfspace duality". It is easy to see that this constraint on $T$ is equivalent to the equation written in the Main Question section, if we take $F(x,y)=(x,-y)$. So the question is equivalent to whether this definition of $T$ is the only way to define a duality between points and halfspaces in such a way that inclusions are preserved.