In this paper we prove a symmetry theorem for the Green function
associated to the heat equation in a certain class of bounded
domains $\Omega\subset\mathbb{R}^{n+1}$. For $T>0$, let
$\Omega_T=\Omega\cap[\mathbb{R}^n\times (0,T)]$ and let $G$ be the
Green function of $\Omega_T$ with pole at
$(0,0)\in\partial_p\Omega_T$. Assume that the adjoint caloric
measure in $\Omega_T$ defined with respect to $(0,0)$, $\hat\omega$,
is absolutely continuous with respect to a certain surface measure,
$\sigma$, on $\partial_p\Omega_T$. Our main result states that if
$$\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}$$
for all $(X,t)\in \partial_p\Omega_T\setminus\{(X,t): t=0\}$ and for
some $\lambda>0$, then
$\partial_p\Omega_T\subseteq\{(X,t):W(X,t)=\lambda\}$ where $W(X,t)$
is the heat kernel and $G=W-\lambda$ in $\Omega_T$. This result has
previously been proven by Lewis and Vogel under stronger assumptions
on $\Omega$.