  Let $G$ be a quadrangulation on an orientable surface and let $g$ be a proper
vertex-$4$-coloring of $G$. A face $F$ of $G$ is said to be a rainbow-face if
all four distinct colors appear on its boundary. A $(c_1,c_2,c_3,c_4)$-face in
$G$ is a rainbow face with colors $c_i$, $i=1,2,3,4$ on the boundary in
clockwise order. We show that the number of $(c_1,c_2,c_3,c_4)$-faces in $G$
equals the number of $(c_4,c_3,c_2,c_1)$-faces. This implies in particular that
the number of rainbow-faces of $G$ is even.

Combinatorics