  We generalize the modern theory of electric polarization to the case of
one-dimensional non-Hermitian systems with line-gapped spectrum. In these
systems, the electronic position operator is non-Hermitian even when projected
into the subspace of states below the energy gap. However, the associated
Wilson-loop operator is biorthogonally unitary in the thermodynamic limit,
thereby leading to real-valued electronic positions that allow for a clean
definition of polarization. Non-Hermitian polarization can be quantized in the
presence of certain symmetries, as for Hermitian insulators. Different from the
latter case, though, in this regime polarization quantization depends also on
the type of energy gap, which can be either real or imaginary, leading to a
richer variety of topological phases. The most counter-intuitive example is the
1D non-Hermitian chain with time-reversal symmetry only, where non-Hermitian
polarization is quantized in presence of an imaginary line-gap. We propose two
specific models to provide numerical evidence supporting our findings.
