  We will show that, contrary to the behavior of the higher order Riesz
transforms studied so far on the atomic Hardy space $\mathcal{H}^1(\mathbb R^n,
\gamma)$, associated with the Ornstein-Uhlenbeck operator with respect to the
$n$-dimensional Gaussian measure $\gamma$, the new Gaussian Riesz transforms
are bounded from $\mathcal{H}^1(\mathbb R^n, \gamma)$ to $L^1(\mathbb R^n,
\gamma)$, for any order and dimension $n$. We will also prove that the
classical Gaussian Riesz transforms of higher order are bounded from an
adequate subspace of $\mathcal{H}^1(\mathbb R^n, \gamma)$ into $L^1(\mathbb
R^n, \gamma)$, extending Bruno's result (J. Fourier Anal. Appl. 25, 4 (2019),
1609--1631) for the first order case.
