$f(10)=?$

Let $f$ be a monic cubic polynomial satisfying $f(x) + f(−x) = 0$, for all real numbers $x$. 

For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x)) = y$. 

Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\big\{1,5,9\big\}$. 

Compute the sum of all possible values of $f(10)$.

HMMT Spring 2021