  Hybrid structures of quantum spin-Hall insulators (QSHIs) and superconductors
(Ss) present a unique opportunity to access dissipationless topological states
of matter, which, however, is frequently hindered by the lack of control over
the spin polarization in QSHIs. We propose a very efficient spin-polarization
mechanism based on the magnetoelectric (Edelstein) effect in superconducting
QSHI structures. It acts akin to the Zeeman splitting in an external magnetic
field, but with an effective $g$-factor of order of 1000, resulting in an
unprecedented spin-splitting effect. It allows a magnetic control of the QSHI/S
hybrids without destroying superconductivity. As an example, we demonstrate a
recurrent crossover from $\Phi_0$ - to $\Phi_0/2$ - periodic oscillations of
the Josephson current in an rf superconducting quantum interference device
($\Phi_0=h/2e$ is the magnetic flux quantum). The predicted period halving is a
striking manifestation of $0-\pi$ Josephson transitions with a superharmonic
$\pi$-periodic current-phase relationship at the transition. Such controllable
$0-\pi$ transitions may offer new perspectives for dissipationless spintronics
and engineering flux qubits.
