Denote $f(1) = a$ and $f(-1) = b$. Choosing $x = y = 1$ and $x = y = -1$ yields the system of equations
$$\begin{cases} 2a^3 + b = a \\ 2b^3 + a = b \end{cases}$$
Adding the equations and cancelling equal terms, we get $2b^3 = -2a^3$. Therefore $b = -a$. Substituting this into the first equation yields $2a^3 - 2a = 0$. So $f(1)$ is one of the numbers $-1$, $0$, $1$. These values can be obtained by the functions $f(x) = -x|x|$, $f(x) = 0$ and $f(x) = x|x|$ respectively, which satisfy the equations.