  Neutron stars are strongly magnetized rotating compact objects. Therefore
they also produce huge electric fields accelerating particles to
ultra-relativistic energies. The simplest magnetic topology is a dipole
traditionally located at the stellar centre. In this paper, we reinvestigate
the consequences of an off-centred rotating magnetic dipole, showing accurate
magnetic field line geometries, the associated spin-down luminosity as well as
the corresponding electromagnetic kick and torque imprinted to the neutron
star. Results are obtained by time dependent numerical simulations of Maxwell
equations in vacuum using pseudo-spectral methods. We compare our results to
known analytical expressions available to lowest order in the parameter
$\epsilon = d/R$, where $d$ is the displacement of the dipole from the stellar
centre and $R$ the neutron star radius. We found good agreement between our
numerical computations and our analytical approximations even for well
off-centred dipoles having large displacements with a sizeable fraction of the
radius, i.e. $\epsilon \lesssim 1$. An explanation for binary neutron star
eccentricity distribution functions is given with an emphasize on highly
eccentric systems as an alternative scenario to traditional binary formation.
