We propose a novel mode of operation for Amplify-and-Forward relays in which the spectra of the relay input and output signals partially overlap. This partial-duplex relaying mode encompasses half-duplex and full-duplex as particular cases. By viewing the partial-duplex relay as a bandwidth-preserving Linear Periodic Time-Varying system, an analysis of the spectral efficiency in the presence of self-interference is developed. In contrast with previous works, self-interference is regarded as a useful information-bearing component rather than simply assimilated to noise. This approach reveals that previous results regarding the impact of self-interference on (full-duplex) relay performance are overly pessimistic. Based on a frequency-domain interpretation of the effect of self-interference, a number of suboptimal decoding architectures at the destination node are also discussed. It is found that the partial-duplex relaying mode may provide an attractive tradeoff between spectral efficiency and receiver complexity.