  MnSb$_{2}$O$_{6}$ is based on the structural chiral $P$321 space group #150
where the magnetic Mn$^{2+}$ moments ($S=5/2$, $L\approx 0$) order
antiferromagnetically at $T_\mathrm{N}=12$ K. Unlike the related iron based
langasite (Ba$_3$NbFe$_3$Si$_2$O$_{14}$) where the low temperature magnetism is
based on a proper helix characterized by a time-even pseudoscalar `magnetic'
chirality, the Mn$^{2+}$ ions in MnSb$_{2}$O$_{6}$ order with a cycloidal
structure at low temperatures, described instead by a time-even vector
`magnetic' polarity. A tilted cycloidal structure has been found [M. Kinoshita
et al. Phys. Rev. Lett. 117, 047201 (2016)] to facilitate ferroelectric
switching under an applied magnetic field. In this work, we apply polarized and
unpolarized neutron diffraction analyzing the magnetic and nuclear structures
in MnSb$_{2}$O$_{6}$ with the aim of understanding this magnetoelectric
coupling. We find no evidence for a helicoidal magnetic structure with one of
the spin envelope axes tilted away from the cycloidal $c$-axis. However, on
application of a magnetic field $\parallel$ $\vec{c}$ the spin rotation plane
can be tilted, giving rise to a cycloid-helix admixture that evolves towards a
distorted helix (zero cycloidal component) for fields great than $\approx$ 2 T.
We propose a mechanism for the previously reported ferroelectric switching
based on coupled structural and magnetic chiralities requiring only an
imbalance of structural chiral domains.
