  By employing resonant X-ray microdiffraction, we image the magnetisation and
magnetic polarity domains of the Y-type hexaferrite
Ba$_{0.5}$Sr$_{1.5}$Mg$_2$Fe$_{12}$O$_{22}$. We show that the magnetic polarity
domain structure can be controlled by both magnetic and electric fields, and
that full inversion of these domains can be achieved simply by reversal of an
applied magnetic field in the absence of an electric field bias. Furthermore,
we demonstrate that the diffraction intensity measured in different X-ray
polarisation channels cannot be reproduced by the accepted model for the polar
magnetic structure, known as the 2-fan transverse conical (TC) model. We
propose a modification to this model, which achieves good quantitative
agreement with all of our data. We show that the deviations from the TC model
are large, and may be the result of an internal magnetic chirality, most likely
inherited from the parent helical (non-polar) phase.
