  A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space
$L^2(\Omega)$ has an orthogonal basis of exponential functions. It is
well-known that in many respects, spectral sets "behave like" sets which can
tile the space by translations. This suggests a conjecture that a product set
$\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are
both spectral sets. We recently proved this in the case when $A$ is an interval
in dimension one. The main result of the present paper is that the conjecture
is true also when $A$ is a convex polygon in two dimensions. We discuss this
result in connection with the conjecture that a convex polytope $\Omega$ is
spectral if and only if it can tile by translations.
