Answer
The function is odd.
Work Step by Step
We have that $\mathbb{E}(-x)=\mathbb{E}(x)$ and $\mathbb{O}(-x)=-\mathbb{O}(x)$.
Let $\Phi=\mathbb{E}\cdot \mathbb{O}$. Then
$$\Phi(-x)=\mathbb{E}(-x)\cdot\mathbb{ O}(-x)=\mathbb{E}(x)\cdot(-\mathbb{O}(x))=-\mathbb{E}(x)\mathbb{O}(x)=-\Phi(x)$$
which means that this function is odd.
