  The Besicovitch covering property (BCP) is known to be one of the fundamental
tools in measure theory, and more generally, a usefull property for numerous
purposes in analysis and geometry. We prove both sufficient and necessary
criteria for the validity of BCP in the first Heisenberg group equipped with a
homogeneous distance. Beyond recovering all previously known results about the
validity or non validity of BCP in this setting, we get simple descriptions of
new large classes of homogeneous distances satisfying BCP. We also obtain a
full characterization of rotationally invariant distances for which BCP holds
in the first Heisenberg group under mild regularity assumptions about their
unit sphere.
