  Measurements of the neutral current cross sections for deep inelastic
scattering in e^- p collisions at HERA with a longitudinally polarised electron
beam are presented. The single-differential cross-sections dsigma/dQ^2,
dsigma/dx and dsigma/dy and the double-differential cross sections in Q^2 and x
are measured in the kinematic region y < 0.9 and Q^2 > 185 GeV^2 for both
positively and negatively polarised electron beams and for each polarisation
state separately. The measurements are based on an integrated luminosity of
169.9 pb^-1 taken with the ZEUS detector in 2005 and 2006 at a centre-of-mass
energy of 318 GeV. The structure functions xF_3 and xF_3^{\gamma Z} are
determined by combining the e^- p results presented in this paper with
previously measured e^+ p neutral current data. The asymmetry parameter A^- is
used to demonstrate the parity violating effects of electroweak interactions at
large spacelike photon virtuality. The measurements agree well with the
predictions of the Standard Model.
