Answer
Intercepts: $(3, 0)$ and $(0,-27)$.
The graph has neither x-axis, y-axis, nor origin symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=x^{3}-27 & y=0^{3}-27\\
x^{3}=27 & y=-27\\
x=3 & \\
&
\end{array}\right]$
Intercepts: $(3, 0)$ and $(0,-27)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=x^{3}-27$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$y=(-x)^{3}-27$
$y=-x^{3}-27$ ... different from initial equation
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y.\ -y=(-x)^{4}-1-y=(-x)^{3}-27$
$y=x^{3}+27$
... different from initial equation