  Complex systems, from the human brain to the global economy, are made of
multiple elements that interact in such ways that the behaviour of the `whole'
often seems to be more than what is readily explainable in terms of the `sum of
the parts.' Our ability to understand and control these systems remains
limited, one reason being that we still don't know how best to describe -- and
quantify -- the higher-order dynamical interactions that characterise their
complexity. To address this limitation, we combine principles from the theories
of Information Decomposition and Integrated Information into what we call
Integrated Information Decomposition, or $\Phi$ID. $\Phi$ID provides a
comprehensive framework to reason about, evaluate, and understand the
information dynamics of complex multivariate systems. $\Phi$ID reveals the
existence of previously unreported modes of collective information flow,
providing tools to express well-known measures of information transfer and
dynamical complexity as aggregates of these modes. Via computational and
empirical examples, we demonstrate that $\Phi$ID extends our explanatory power
beyond traditional causal discovery methods -- with profound implications for
the study of complex systems across disciplines.
