We present high resolution blue spectroscopy of an almost complete sample of optical counterparts to massive X-ray binaries in the Large Magellanic Cloud (LMC) and derive their spectral classification. We find an spectral type B0II for the optical counterpart to RX J0532.5-6551, confirming it as the first wind-fed massive X-ray binary in the LMC. We also confirm the Be nature of the proposed counterpart to RX J0535.0-6700. The proposed optical counterpart to RX J0531.5-6518 is a B2V star with signs of emission in the Balmer lines. In total, we give accurate spectral types for 14 counterparts. We find that the overall observed population of massive X-ray binaries in the LMC has a distribution not very different from the observed Galactic population and we discuss different selection effects affecting our knowledge of this population. The spectral distribution of the Be/X-ray binary population is also rather similar to the Galactic one. This distribution implies that Be/X-ray binaries must have preferentially formed from moderately massive binaries undergoing semi-conservative evolution. The observation of several Be/X-ray binaries with large eccentricities implies then the existence of supernova kicks.