  Information-theoretic quantities reveal dependencies among variables in the
structure of joint, marginal, and conditional entropies, but leave some
fundamentally different systems indistinguishable. Furthermore, there is no
consensus on how to construct and interpret a higher-order generalisation of
mutual information (MI). In this manuscript, we show that a recently proposed
model-free definition of higher-order interactions amongst binary variables
(MFIs), like mutual information, is a M\"obius inversion on a Boolean algebra,
but of surprisal instead of entropy. This gives an information-theoretic
interpretation to the MFIs, and by extension to Ising interactions. We study
the dual objects to MI and MFIs on the order-reversed lattice, and find that
dual MI is related to the previously studied differential mutual information,
while dual interactions (outeractions) are interactions with respect to a
different background state. Unlike mutual information, in- and outeractions
uniquely identify all six 2-input logic gates, the dy- and triadic
distributions, and different causal dynamics that are identical in terms of
their Shannon-information content.
