Receiving an asialoglycoprotein, and delivering the asialoglycoprotein into the cell via endocytosis. An asialoglycoprotein is a plasma glycoproteins from which the terminal sialic acid residue on their complex carbohydrate groups has been removed. The asialoglycoprotein receptor recognizes the terminal galactose and N-acetylgalactosamine units of the asialoglycoprotein, the receptor-ligand complex is internalized and transported to a sorting organelle where disassociation occurs before the receptor is recycled to the cell membrane.