  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.
