  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.
