Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space)
What is a good geometric interpretation of this relationship? Obviously we could say that $y - x$ and $F(y) - F(x)$ maintains a less than $90$ deg angle, but can we say more about that?
Does $\langle y - x, F(y) - F(x)\rangle \leq 0$ also define a monotone operator? Notice we cannot just pull out negative signs in the original inequality and reverse the sign this way.
Part 2 is easier to answer: no, the way monotone operators are defined in functional analysis, $-F$ is not in general monotone when $F$ is. (This is unlike the concept of monotonicity in real analysis). Monotone operators correspond to (non-strictly) increasing functions. The reverse inequality defines dissipative operators.
Part 1, geometric interpretation beyond "at most 90 degree rotation". If the graph of an increasing function is rotated by $45$ degrees clockwise, the result is the graph of a $1$-Lipschitz function. (And conversely.) This property also holds for monotone operators, in the following form: consider the graph of $F$ as a subset of $X\times X$ and apply the linear transformation $(x,y)\mapsto (x+y,y-x)$ in $X\times X$. The image of the graph of $F$ under this transformation is the graph of a $1$-Lipschitz function. Indeed,the inequality $$ \|(y_1-x_1)-(y_2 -x_2)\| \le \|(y_1 +x_1)-(y_2 +x_2)\| $$ is equivalent to $\langle y_1-y_2,x_1-x_2\rangle \ge 0$.