  For an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to a
hyperplane and with positive part $\Omega_+$, we consider the Neumann/Dirichlet
Laplacians $-\Delta_{N/D,\Omega}$ and $-\Delta_{N/D,\Omega_+}$. Given a Borel
function $\Phi$ on $[0,\infty)$ we apply the spectral functional calculus and
consider the pairs of operators $\Phi(-\Delta_{N,\Omega})$ and
$\Phi(-\Delta_{N,\Omega_+})$, or $\Phi(-\Delta_{D,\Omega})$ and
$\Phi(-\Delta_{D,\Omega_+})$. We prove relations between the integral kernels
for the operators in these pairs, which in particular cases of
$\Omega_+=\mathbb{R}^{d-1}\times(0,\infty)$ and $\Phi_{t}(u)=\exp(-tu)$, $u
\geq 0$, $t>0$, were known as reflection principles for the Neumann/Dirichlet
heat kernels. These relations are then generalized to the context of symmetry
with respect to a finite number of mutually orthogonal hyperplanes.
