This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that
$$ (1): \quad f(x) = -f(-x) $$ and $$ (2): \quad f(x_1 + x) = -f(x_1 - x) $$ for all $x$.
Then, the vague question I'd like to answer is
What else can we conclude about this function?
It seems maybe I can conclude it is periodic, with period $2x_1$ (in general $2\left|x_1 - x_0\right|$): given the values of $f$ on $(0,x_1)$, $(1)$ determines the values on $(-x_1,0)$, then this and $(2)$ determine the values on $(x_1,3x_1)$, then this and $(1)$ determine the values of $f$ on $(-3x_1,-x_1)$, this and $(2)$ determine the values on $(3x_1,5x_1)$, and so forth.
Is there anything else to say here?