  We investigate spatial reflection and associated nonlocal order in spin chain
quantum systems. The proposed string order parameters, e.g., reflected via
operations of the spatial reflection or combinations of it with spin
reflection, are able to characterize a variety of physical systems and allow us
to gain renewed insights to the statistical mechanism underlying phenomena such
as the Haldane gap and quantum phase transitions. Besides revealing further the
potential application of the generalized parity symmetry in numerical
algorithm, we build an explicit scheme to determine the symmetry and the
related string order for matrix product states so that one can construct ansatz
models with presumed properties.
