  We study several fundamental harmonic analysis operators in the
multi-dimensional context of the Dunkl harmonic oscillator and the underlying
group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit
negative values of the multiplicity functions. Our investigations include
maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and
multipliers of Laplace and Laplace-Stieltjes type. By means of the general
Calder\'on-Zygmund theory we prove that these operators are bounded on weighted
$L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$.
We also obtain similar results for analogous set of operators in the closely
related multi-dimensional Laguerre-symmetrized framework. The latter emerges
from a symmetrization procedure proposed recently by the first two authors. As
a by-product of the main developments we get some new results in the
multi-dimensional Laguerre function setting of convolution type.
