  The magnetic field induced rearrangement of the cycloidal spin structure in
ferroelectric mono-domain single crystals of the room-temperature multiferroic
BiFeO$_3$ is studied using small-angle neutron scattering (SANS). The cycloid
propagation vectors are observed to rotate when magnetic fields applied
perpendicular to the rhombohedral (polar) axis exceed a pinning threshold value
of $\sim$5\,T. In light of these experimental results, a phenomenological model
is proposed that captures the rearrangement of the cycloidal domains, and we
revisit the microscopic origin of the magnetoelectric effect. A new coupling
between the magnetic anisotropy and the polarization is proposed that explains
the recently discovered magnetoelectric polarization to the rhombohedral axis.
