  Studying the primordial non-Gaussianity of inflationary perturbations is
crucial for testing the inflation paradigm of the early universe. In this work,
we conduct a comprehensive analysis of the angular bispectrum and trispectrum
of scalar-induced gravitational waves (SIGWs) in the presence of local-type
primordial non-Gaussianity parameterized by $f_\mathrm{NL}$ and
$g_\mathrm{NL}$, deriving their semi-analytical formulae for the first time.
Our findings indicate that it is the presence of primordial non-Gaussianity
that leads to a non-Gaussian SIGW background, suggesting that the angular
bispectrum and trispectrum of SIGWs could serve as probes of the primordial
non-Gaussianity. Our numerical results further illustrate that $f_\mathrm{NL}$
and $g_\mathrm{NL}$ exert significant impacts on the spectral amplitudes,
potentially reaching up to $10^{-5}$ for the former and $10^{-8}$ for the
latter. In particular, we demonstrate that the angular bispectrum and
trispectrum exhibit characteristic dependence on the angular multipoles and
frequency bands. They hold potentials to be measured by gravitational-wave
detectors that may advance our understanding of the origin of the universe.
