  In our previous work, we proposed an algorithm to transform the metric of an
isolated matter source in the multipolar post-Minkowskian approximation in
harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this
algorithm at linear order and for specific quadratic interactions known as
quadratic tail terms. In the present work, we extend this analysis to quadratic
interactions associated with the coupling of two mass quadrupole moments,
including both instantaneous and hereditary terms. Our main result is the
derivation of the metric in Newman-Unti and Bondi gauges with complete
quadrupole-quadrupole interactions. We rederive the displacement memory effect
and provide expressions for all Bondi aspects and dressed Bondi aspects
relevant to the study of leading and subleading memory effects. Then we obtain
the Newman-Penrose charges, the BMS charges as well as the second and third
order celestial charges defined from the known second order and novel third
order dressed Bondi aspects for mass monopole-quadrupole and
quadrupole-quadrupole interactions.
