Why Fourier transform and Stockwell-transform retain the absolute phase information of one signal?

Hello friends am studying the topic of signal processing and the Fourier transform and the s-transform and in most books as for example "Time-Frequency Signal Analysis and Processing. 2nd" of Boashash in section 5.11.2.3 says that these transforms retain the absolute phase information of the signal and I do not understand that it mean with this concept about retain the absolute phase information of the signal and why this is important?.

It simply means that for a signal: $$x(t) = a(t) e^{j\phi(t)}$$ that the phase information $\phi(t)$ is completely preserved in the transform domain.
Time-frequency representations that use a quadratic form do not necessarily have this property: $$x(t+\tau)x^*(t-\tau) = a(t+\tau)a(t-\tau)e^{j(\phi(t+\tau) - \phi(t-\tau))}$$ so if $\phi$ is of the form: $$\phi(t) = f(t) + C$$ where $C$ is a constant, then $C$ is not preserved in $x(t+\tau)x^*(t-\tau)$.