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Nonreciprocity means that the transmission of a signal depends on its direction of propagation. Despite vastly different platforms and underlying working principles, the realisations of nonreciprocal transport in linear, time-independent systems rely on Aharonov-Bohm interference among several pathways and require breaking time-reversal symmetry. Here we extend the notion of nonreciprocity to unidirectional bosonic transport in systems with a time-reversal symmetric Hamiltonian by exploiting interference between beamsplitter (excitation preserving) and two-mode-squeezing (excitation non-preserving) interactions. In contrast to standard nonreciprocity, this unidirectional transport manifests when the mode quadratures are resolved with respect to an external reference phase. Hence we dub this phenomenon quadrature nonreciprocity. First, we experimentally demonstrate it in the minimal system of two coupled nanomechanical modes orchestrated by optomechanical interactions. Next, we develop a theoretical framework to characterise the class of networks exhibiting quadrature nonreciprocity based on features of their particle-hole graphs. In addition to unidirectionality, these networks can exhibit an even-odd pairing between collective quadratures, which we confirm experimentally in a four-mode system, and an exponential end-to-end gain in the case of arrays of cavities. Our work opens up new avenues for signal routing and quantum-limited amplification in bosonic systems.