  In recent years, temperature scales in cool supergiants (CSGs) have been
disputed, and the possibility that spectral types (SpTs) do not depend
primarily on temperature has been raised. We explore the relations between
different observed parameters and the capability of deriving accurate intrinsic
stellar parameters from them through the analysis of the largest spectroscopic
sample of CSGs to date from SMC and LMC. We explore possible correlations
between different observational parameters, also making use of near- and
mid-infrared colours and literature on photometric variability. Direct
comparison between the behaviour of atomic lines (Fe I, Ti I, and Ca II) in the
observed spectra and synthetic atmospheric models provides compelling evidence
that effective temperature is the prime underlying variable driving the SpT
sequence in CSGs. However, there is a clear correlation between SpT and
luminosity, with later ones tending to correspond to more luminous stars with
heavier mass loss. The population of CSGs in the SMC is characterised by a
higher degree of spectral variability, early spectral types (centred on type
K1) and low mass-loss rates (at least as measured by dust-sensitive
mid-infrared colours). The population in the LMC displays less spectroscopic
variability and later spectral types. The distribution of spectral types is not
single-peaked. Instead, the brightest CSGs have a significantly different
distribution from less luminous objects, presenting mostly M subtypes (centred
on M2), and increasing mass-loss rates for later types. In conclusion, the
observed properties of CSGs in the SMC and the LMC cannot be described
correctly by standard evolutionary models. The very strong correlation between
spectral type and bolometric luminosity, supported by all data from the Milky
Way, cannot be reproduced at all by current evolutionary tracks.
