  Quantum phases with different orders exist with or without breaking the
symmetry of the system. Recently, a classification of gapped quantum phases
which do not break time reversal, parity or on-site unitary symmetry has been
given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B
\textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry
protected topological (SPT) phases are labeled by the projective
representations of the symmetry group which can be viewed as a symmetry
fractionalization. In this paper, we extend the classification of 1D gapped
phases by considering SPT phases with combined time reversal, parity, and/or
on-site unitary symmetries and also the possibility of symmetry breaking. We
clarify how symmetry fractionalizes with combined symmetries and also how
symmetry fractionalization coexists with symmetry breaking.
  In this way, we obtain a complete classification of gapped quantum phases in
1D spin systems. We find that in general, symmetry fractionalization, symmetry
breaking and long range entanglement(present in 2 or higher dimensions)
represent three main mechanisms to generate a very rich set of gapped quantum
phases. As an application of our classification, we study the possible SPT
phases in 1D fermionic systems, which can be mapped to spin systems by
Jordan-Wigner transformation.
