For K-shell photoionization of neon dimers, we report Ne 1s photoelectron angular distributions for Ne(2+) + Ne(+) and Ne(+) + Ne(+) channels exhibiting quite different patterns. Noninversion-symmetric patterns of the former obtained by the fast interatomic Coulombic decay of Auger final states show direct evidence of core-hole localization. Dipolar patterns of the latter obtained by the slow radiative decay of the other Auger final states clearly show that the radiative process is slow enough to allow dicationic dimers to rotate many times before fragmentation.