Answer
Intercepts: $(0,0)$.
The graph has origin symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=\sqrt[3]{x} & y=\sqrt[3]{0}\\
x=0 & y=0\\
&
\end{array}\right]$
The only intercept is $(0,0)$
Test x-axis symmetry: Replace $y$ with $-y$
$-y=\sqrt[3]{x}$
... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$y=\sqrt[3]{-x}$
$y=-\sqrt[3]{x}$ ... different from initial equation
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=\sqrt[3]{-x}$
$-y=-\sqrt[3]{x}$
$y=\sqrt[3]{x}$
...same as initial equation