  We report on spherical neutron polarimetry and unpolarized neutron
diffraction in zero magnetic field as well as flipping ratio and static
magnetization measurements in high magnetic fields on the multiferroic square
lattice antiferromagnet Ba$_2$CoGe$_2$O$_7$. We found that in zero magnetic
field the magnetic space group is $Cm'm2'$ with sublattice magnetization
parallel to the [100] axis of this orthorhombic setting. The spin canting has
been found to be smaller than $0.2^\circ$ in the ground state. This assignment
is in agreement with the field-induced changes of the magnetic domain structure
below 40 mT as resolved by spherical neutron polarimetry. The magnitude of the
ordered moment has been precisely determined. Above the magnetic ordering
temperature short-range magnetic fluctuations are observed. Based on the
high-field magnetization data, we refined the parameters of the recently
proposed microscopic spin model describing the multiferroic phase of
Ba$_2$CoGe$_2$O$_7$.
