  The spectral fluctuations of complex quantum systems, in appropriate limit,
are known to be consistent with that obtained from random matrices. However,
this relation between the spectral fluctuations of physical systems and random
matrices is valid only if the spectra are desymmetrized. This implies that the
fluctuation properties of the spectra are affected by the discrete symmetries
of the system. In this work, it is shown that in the chaotic limit the
fluctuation characteristics and symmetry structure for any arbitrary sequence
of measured or computed levels can be inferred from its higher-order spectral
statistics without desymmetrization. In particular, we consider a spectrum
composed of $k>0$ independent level sequences with each sequence having the
same level density. The $k$-th order spacing ratio distribution of such a
composite spectrum is identical to its nearest neighbor counterpart with
modified Dyson index $k$. This is demonstrated for the spectra obtained from
random matrices, quantum billiards, spin chains and experimentally measured
nuclear resonances with disparate symmetry features.
