We provide necessary and sufficient structural conditions for multistationarity and oscillations in aggregate monotone systems, defined as the interconnection of stable monotone components. Our classification is based on the presence of exclusively positive or exclusively negative cycles in the system aggregate graph, whose nodes are the monotone subsystems. The results presented here scale up the structural classification of oscillatory and multistationary behaviors in sign-definite biochemical networks previously proposed by the authors. Models of biomolecular systems are discussed to demonstrate the applicability of our classification.