  Observations of galaxy clustering are made in redshift space, which results
in distortions to the underlying isotropic distribution of galaxies. These
redshift-space distortions (RSD) not only degrade important features of the
matter density field, such as the baryonic acoustic oscillation (BAO) peaks,
but also pose challenges for the theoretical modelling of observational probes.
Here we introduce an iterative nonlinear reconstruction algorithm to remove RSD
effects from galaxy clustering measurements, and assess its performance by
using mock galaxy catalogues. The new method is found to be able to recover the
real-space galaxy correlation function with an accuracy of $\sim1\%$, and
restore the quadrupole accurately to $0$, on scales $s\gtrsim20\Mpch$. It also
leads to an improvement in the reconstruction of the initial density field,
which could help to accurately locate the BAO peaks. An `internal calibration'
scheme is proposed to determine the values of cosmological parameters as a part
of the reconstruction process, and possibilities to break parameter
degeneracies are discussed. RSD reconstruction can offer a potential way to
simultaneously extract the cosmological parameters, initial density field,
real-space galaxy positions and large-scale peculiar velocity field (of the
real Universe), making it an alternative to standard perturbative approaches in
galaxy clustering analysis, bypassing the need for RSD modelling.
