  In recent years, motivated by computational purposes, the singular value and
spectral features of the symmetrization of Toeplitz matrices generated by a
Lebesgue integrable function have been studied. Indeed, under the assumptions
that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the
spectral and singular value distribution of the matrix-sequence
$\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix-size, $Y_n$ is
the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$.
In this note, we consider the multilevel Toeplitz matrix $T_{\bf n}[f]$
generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying
the matrix-size, and we prove spectral and singular value distribution results
for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$
being the corresponding tensorization of the anti-identity matrix.
