We present ESO-NTT low resolution spectroscopy of the neutron star X-ray transient Cen X-4 in quiescence over a complete orbital cycle. Our data reveal the presence of a K3-7 V companion which contributes 63% to the 5600-6900 Å flux and orbits the neutron star with a velocity semi-amplitude of K_2=145.8± 1.0 km s-1. This, combined with a previous determination of the inclination angle and mass ratio, yields a neutron star and companion mass of M_1=1.5±1.0~M&sun; and M_2=0.31±0.27~M&sun;, respectively. The mass donor is thus undermassive for the inferred spectral type indicating it is probably evolved, in agreement with previous studies. Doppler tomography of the Hα line shows prominent emission located on the companion and a slightly asymmetric accretion disc distribution similar to that seen in systems with precessing eccentric discs. Strong Hα emission from the companion can be explained by X-ray irradiation from the primary. No evidence is found for a hot spot in Hα, whereas one is revealed via Doppler tomography of the HeI lines. This can be interpreted as the hot spot and outer regions of the disc being at a higher temperature than in other systems.