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 d σ/dQ ², d σ/dx and d σ/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 F3 and xF ₃ ^{γ 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.