  It is generally believed that it is not possible to have a four dimensional
differential calculus in $\kappa$-Minkowski spacetime, with $\kappa$-Poincar\'e
relativistic symmetries, covariant under ($\kappa$-deformed) Lorentz
transformations. Thus, one usually introduces a fifth differential form, whose
physical interpretation is still challenging, and defines a covariant five
dimensional calculus. Nevertheless, the four dimensional calculus is at the
basis of several works based on $\kappa$-Minkowski/$\kappa$-Poincar\'e
framework that led to meaningful insights on its physical interpretation and
phenomenological implications. We here revisit the argument against the
covariance of the four dimensional calculus, and find that it depends crucially
on an incomplete characterization of Lorentz transformations in this framework.
In particular, we understand that this is due to a feature, still uncovered at
the time, that turns out to be fundamental for the consistency of the
relativistic framework: the noncommutativity of the Lorentz transformation
parameters. Once this is taken into account, the four dimensional calculus is
found to be fully Lorentz covariant. The result we obtain extends naturally to
the whole $\kappa$-Poincar\'e algebra of transformations, showing the close
relation between its relativistic nature and the properties of the differential
calculus.
