  We calculate the beating of $h/2e$ and $h/e$ periodic oscillations of the
flux-dependent critical supercurrent $I_c(\Phi)$ through a quantum spin-Hall
insulator between two superconducting electrodes. A conducting pathway along
the superconductor connects the helical edge channels via a non-helical
channel, allowing an electron incident on the superconductor along one edge to
be Andreev reflected along the opposite edge. In the limit of small Andreev
reflection probability the resulting even-odd effect is described by
$I_c\propto|\cos(e\Phi/\hbar)+f|$, with $|f|\ll 1$ proportional to the
probability for phase-coherent inter-edge transmission. Because the sign of $f$
depends on microscopic details, a sample-dependent inversion of the alternation
of large and small peaks is a distinctive feature of the beating mechanism for
the even-odd effect.
