It's insensitive to individual phase of audible frequencies, but the concept of phase itself is very important in audio in the form of time delays/echoes, if yo

It's insensitive to individual phase of audible frequencies, but the concept of phase itself is very important in audio in the form of time delays/echoes, if you are dealing with long sample lengths. See my sibling comment explaining how translation corresponds to ramping phase shift/"fast-forwarding" each frequencies such that the shifted distance are the same across the spectrum.