  We establish the stochastic comparison principles, including moment
comparison principle as a special case, for solutions to the following
nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial
}{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x)) \:\dot{M}(t,x),
\] where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in
time and colored in space, and $\rho$ is a Lipschitz continuous function that
vanishes at zero. These results are obtained for rough initial data and under
Dalang's condition, namely,
$\int_{\mathbb{R}^d}(1+|\xi|^2)^{-1}\hat{f}(\text{d} \xi)<\infty$, where
$\hat{f}$ is the spectral measure of the noise. We establish the comparison
principles by comparing either the diffusion coefficient $\rho$ or the
correlation function of the noise $f$. As corollaries, we obtain Slepian's
inequality for SPDEs and SDEs.
