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Temporal irreversibility, often referred to as the arrow of time, is a
fundamental concept in statistical mechanics. Markers of irreversibility also
provide a powerful characterisation of information processing in biological
systems. However, current approaches tend to describe temporal irreversibility
in terms of a single scalar quantity, without disentangling the underlying
dynamics that contribute to irreversibility. Here we propose a broadly
applicable information-theoretic framework to characterise the arrow of time in
multivariate time series, which yields qualitatively different types of
irreversible information dynamics. This multidimensional characterisation
reveals previously unreported high-order modes of irreversibility, and
establishes a formal connection between recent heuristic markers of temporal
irreversibility and metrics of information processing. We demonstrate the
prevalence of high-order irreversibility in the hyperactive regime of a
biophysical model of brain dynamics, showing that our framework is both
theoretically principled and empirically useful. This work challenges the view
of the arrow of time as a monolithic entity, enhancing both our theoretical
understanding of irreversibility and our ability to detect it in practical
applications.