In performing FFTs and looking at spectra for real signals (say, taken from a sensor) we ignore the negative frequencies. I understand that the negative frequencies often reflect the positive frequencies because their imaginary components are complex conjugates. Is this only the case for real signals? Would we be able to notice a difference if imaginary components of a signal did not cancel out (as in hearing an audio signal)?
A strictly real vector in one domain implies a complex conjugate mirrored vector in the other domain, and vice versa, for any FFT.
The "negative" frequency results of an FFT are very commonly used in SDR (Software Defined Radio), where the signal of interest might consist of only negative frequencies (LSB or lower sideband SSB is one obvious case) in a baseband IQ stream (after quadrature downconversion from strictly real higher frequency RF). Or the SDR software may display a wideband spectrum showing signals both above and below the IQ baseband center frequency, which are displayed from an FFTs negative frequency bins thru positive frequency bins.
Some USB SDR devices (Funcube, et.al.) communicate the complex IQ data stream as stereo audio inputs to the computer, so one might actually be able to listen to the complex IQ data as a stereo left and right pair of audio (after appropriate sample rate conversion, if needed).