  We obtain several new characterizations of splayedness for divisors: a
Leibniz property for ideals of singularity subschemes, the vanishing of a
`splayedness' module, and the requirements that certain natural morphisms of
modules and sheaves of logarithmic derivations and logarithmic differentials be
isomorphisms. We also consider the effect of splayedness on the Chern classes
of sheaves of differential forms with logarithmic poles along splayed divisors,
as well as on the Chern-Schwartz-MacPherson classes of the complements of these
divisors. A postulated relation between these different notions of Chern class
leads to a conjectural identity for Chern-Schwartz-MacPherson classes of
splayed divisors and subvarieties, which we are able to verify in several
template situations.
