Motivation of time-frequency analysis
What a tricky question to overlook. Indeed I'm one of those who would immedieately press that Fourier transforms do lose time localization of the events as the comments stated. Yet it's certainly (mathematically and practically) true that any (transformable) signal waveform is exactly preserved under this reversible transform including all of its time distribution information as well. Your example of the reversed chirp case clearly demonstrates this. This needs an answer. At least to clarify why then there is the well accepted idea that FT do not preserve time localization of events?

Now, the idea that the Fourier transforms are losing time localization of the signals come from the observation that its bases are of infinite extent sinusoidals. And a sine wave of infinite extent and constant amplitude will not have a time localization. On the other hand it has a perfect frequency localization, being a pair of impulses at an exact frequency. The bases of wavelets, for example, are partly local in both domains.

This is the essence of time-frequency analysis. A base that's exactly local in frequency will loose all time localization and a base that's exactly local in time will loose any frequency localization. And those in between are the transforms that provide a compromise between exact localization and no localization at all. 

A consequence of the fact that Fourier bases are of infininte extent is the following: Assume part of a signal contains a short duration high frequency spike, and the remaining parts are of quite still low frequency variations. When a Fourier decomposition or synthesis is used to analyse or construct such a signal, the high frequency spike (which is higly local in time) is to be created with high frequency sine bases that not only exist at the position of the spike but also exist along all the rest of the signal. Because those sines will extend from the beginning to the end of the whole signal duration. This creates the problem that local processing is quite inefficient with Fourier bases for transient signals. And transient signals have the prime importance in many mathematical, scientific and engineering fields. 

Wavelets are kind of the optimal (probably in some sense?) transforms that provide a maximum amount of simultaneous time-frequency localizations (resolutions). And this is clearly apparent from the wave-packet (or the gaussian, or the maxican hat, or a shapeless daubechies...) shape  of its bases. 
Wavelets therefore can provide an improved solution the problem associated with the previous paragraph.

A spectogram is partly providing similar information to a wavelet time-frequency analysis, and gives you frequency contents of the signal over its time duration.  

Note that you can still go with Fourier analysis for transient cases as well but that requires what Peter K. referred to as a codswallop work...