  Real-world multi-layer networks feature nontrivial dependencies among links
of different layers. Here we argue that, if links are directed, dependencies
are twofold. Besides the ordinary tendency of links of different layers to
align as the result of `multiplexity', there is also a tendency to anti-align
as the result of what we call `multireciprocity', i.e. the fact that links in
one layer can be reciprocated by \emph{opposite} links in a different layer.
Multireciprocity generalizes the scalar definition of single-layer reciprocity
to that of a square matrix involving all pairs of layers. We introduce
multiplexity and multireciprocity matrices for both binary and weighted
multiplexes and validate their statistical significance against maximum-entropy
null models that filter out the effects of node heterogeneity. We then perform
a detailed empirical analysis of the World Trade Multiplex (WTM), representing
the import-export relationships between world countries in different
commodities. We show that the WTM exhibits strong multiplexity and
multireciprocity, an effect which is however largely encoded into the degree or
strength sequences of individual layers. The residual effects are still
significant and allow to classify pairs of commodities according to their
tendency to be traded together in the same direction and/or in opposite ones.
We also find that the multireciprocity of the WTM is significantly lower than
the usual reciprocity measured on the aggregate network. Moreover, layers with
low (high) internal reciprocity are embedded within sets of layers with
comparably low (high) mutual multireciprocity. This suggests that, in the WTM,
reciprocity is inherent to groups of related commodities rather than to
individual commodities. We discuss the implications for international trade
research focusing on product taxonomies, the product space, and
fitness/complexity metrics.
