Most information dynamics and statistical causal analysis frameworks rely on the common intuition that causal interactions are intrinsically pairwise -- every cause variable has an associated effect variable, so that a causal arrow can be drawn between them. However, analyses that depict interdependencies as directed graphs fail to discriminate the rich variety of modes of information flow that can coexist within a system. This, in turn, creates problems with attempts to operationalise the concepts of dynamical complexity or `integrated information. To address this shortcoming, we combine concepts of partial information decomposition and integrated information, and obtain what we call Integrated Information Decomposition, or $Phi$ID. We show how $Phi$ID paves the way for more detailed analyses of interdependencies in multivariate time series, and sheds light on collective modes of information dynamics that have not been reported before. Additionally, $Phi$ID reveals that what is typically referred to as integration is actually an aggregate of several heterogeneous phenomena. Furthermore, $Phi$ID can be used to formulate new, tailored measures of integrated information, as well as to understand and alleviate the limitations of existing measures.
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