The associated production of Higgs and W bosons via vector-boson fusion is highly sensitive to the relative sign of the Higgs boson couplings to W and Z bosons. In this Letter, two searches for this process are presented, using 140 fb^{-1} of proton-proton collision data at sqrt[s]=13 TeV recorded by the ATLAS detector at the LHC. The first search targets scenarios with opposite-sign couplings of the W and Z bosons to the Higgs boson, while the second targets standard model-like scenarios …