  We discuss in detail and from the geometrical point of view the issues of
gauge invariance and Lorentz covariance raised by the approach proposed
recently by Chen et al. to the proton spin decomposition. We show that the
gauge invariance of this approach follows from a mechanism similar to the one
used in the famous Stueckelberg trick. Stressing the fact that the Lorentz
symmetry does not force the gauge potential to transform as a Lorentz
four-vector, we show that the Chen et al. approach is Lorentz covariant
provided that one uses the suitable Lorentz transformation law. We also make an
attempt to summarize the present situation concerning the proton spin
decomposition. We argue that the ongoing debates concern essentially the
physical interpretation and are due to the plurality of the adopted pictures.
We discuss these different pictures and propose a pragmatic point of view.
