A central question in neuroscience concerns the relationship between consciousness and its physical substrate. Here, we argue that a richer characterisation of consciousness can be obtained by viewing it not as a monolithic construct, but rather as constituted of distinct information-theoretic elements. In other words, we propose a shift from quantification of consciousness - viewed as integrated information - to its decomposition. Through this approach, termed Integrated Information Decomposition (ΦID), we lay out a formal argument that whether the consciousness of a given system is an emergent phenomenon depends on its information-theoretic composition - thus providing a principled answer to the long-standing dispute on the relationship between consciousness and emergence. Furthermore, we show that two organisms may attain the same amount of integrated information, yet differ in its information-theoretic composition. Building on ΦID’s revised understanding of integrated information, termed ΦR, we also introduce the notion of ΦR-ing rate to quantify how efficiently an entity uses information for conscious processing. A combination of ΦR and ΦR-ing rate may provide an important way to compare the neural basis of different aspects of consciousness. Thus, decomposition of consciousness enables us to identify qualitatively different ‘modes of consciousness,’ which establish a common space for mapping the phenomenology of different conscious states. We outline both theoretical and empirical avenues to carry out such mapping between phenomenology and information-theoretic modes, starting from a central feature of everyday consciousness: selfhood. Overall, Integrated Information Decomposition yields rich new ways to explore the relationship between information, consciousness, and its emergence from neural dynamics.