  Quantum multifractality is a fundamental property of systems such as
non-interacting disordered systems at an Anderson transition and many-body
systems in Hilbert space. Here we discuss the origin of the presence or absence
of a fundamental symmetry related to this property. The anomalous multifractal
dimension $\Delta_q$ is used to characterize the structure of quantum states in
such systems. Although the multifractal symmetry relation
\mbox{$\Delta_q=\Delta_{1-q}$} is universally fulfilled in many known systems,
recently some important examples have emerged where it does not hold. We show
that this is the result of two different mechanisms. The first one was already
known and is related to Gaussian fluctuations well described by random matrix
theory. The second one, not previously explored, is related to the presence of
an algebraically localized envelope. While the effect of Gaussian fluctuations
can be removed by coarse graining, the second mechanism is robust to such a
procedure. We illustrate the violation of the symmetry due to algebraic
localization on two systems of very different nature, a 1D Floquet critical
system and a model corresponding to Anderson localization on random graphs.
