  We present a general framework for a comparative theory of variability
measures, with a particular focus on the recently introduced one-parameter
families of inter-Expected Shortfall differences and inter-expectile
differences, that are explored in detail and compared with the widely known and
applied inter-quantile differences. From the mathematical point of view, our
main result is a characterization of symmetric and comonotonic variability
measures as mixtures of inter-Expected Shortfall differences, under a few
additional technical conditions. Further, we study the stochastic orders
induced by the pointwise comparison of inter-Expected Shortfall and
inter-expectile differences, and discuss their relationship with the dilation
order. From the statistical point of view, we establish asymptotic consistency
and normality of the natural estimators and provide a rule of the thumb for
cross-comparisons. Finally, we study the empirical behaviour of the considered
classes of variability measures on the S&P 500 Index under various economic
regimes, and explore the comparability of different time series according to
the introduced stochastic orders.
