In recent years, many natural systems are identified to demonstrate multiscale organization, which can be quantitatively described within the framework of the multifractal formalism and the related singularity spectra. The mathematical cascade-like models are characterized by a singularity spectrum whose shape is typically similar to an inverted symmetric parabola, whereas for empirical data we usually obtain asymmetrical spectra. This presentation shows that asymmetry in the singularity spectrum provides important information about nonuniform organization of the fractal time series. We show that financial time series are characterized mainly by the left-sided spectrum asymmetry that can be interpreted as an indication of the multifractal organization of large events, while the dynamics of small  ones is less complex and can be regarded as a kind of noise. On the other hand, the right-sided spectrum asymmetry occurs less frequently and is less intuitive. It has been identified, for example, in the intertransaction intervals. This shape suggests the multifractal organization of small fluctuations and monofractality of the large ones. We also present models that are able to generate the signals with a given type of asymmetry in their singularity spectra, which support our interpretation of the asymmetry phenomenon.

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