  Let $W$ be a finite reflection group associated with a root system $R$ in
$\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open
subset $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from
$W$. Let $\Omega_+=\Omega\cap C_+$ be the positive part of $\Omega$. We define
a family $\{-\Delta_{\eta}^+\}$ of self-adjoint extensions of the Laplacian
$-\Delta_{\Omega_+}$, labeled by homomorphisms $\eta\colon W\to \{1,-1\}$. In
the construction of these $\eta$-Laplacians $\eta$-symmetrization of functions
on $\Omega$ is involved. The Neumann Laplacian $-\Delta_{N,\Omega_+}$ is
included and corresponds to $\eta\equiv1$. If $H^{1}(\Omega)=H^{1}_0(\Omega)$,
then the Dirichlet Laplacian $-\Delta_{D,\Omega_+}$ is either included and
corresponds to $\eta={\rm sgn}$; otherwise the Dirichlet Laplacian is
considered separately. Applying the spectral functional calculus we consider
the pairs of operators $\Psi(-\Delta_{N,\Omega})$ and $\Psi(-\Delta_{\eta}^+)$,
or $\Psi(-\Delta_{D,\Omega})$ and $\Psi(-\Delta_{D,\Omega_+})$, where $\Psi$ is
a Borel function on $[0,\infty)$. We prove relations between the integral
kernels for the operators in these pairs, which are given in terms of
symmetries governed by $W$.
