  The family of skew-symmetric distributions is a wide set of probability
density functions obtained by combining in a suitable form a few components
which are selectable quite freely provided some simple requirements are
satisfied. Intense recent work has produced several results for specific
sub-families of this construction, but much less is known in general terms. The
present paper explores some questions within this framework, and provides
conditions on the above-mentioned components to ensure that the final
distribution enjoys specific properties.
