  We study the properties of tilting modules in the context of properly
stratified algebras. In particular, we answer the question when the Ringel dual
of a properly stratified algebra is properly stratified itself, and show that
the class of properly stratified algebras for which the characteristic tilting
and cotilting modules coincide is closed under taking the Ringel dual. Studying
stratified algebras, whose Ringel dual is properly stratified, we discover a
new Ringel-type duality for such algebras, which we call the two-step duality.
This duality arises from the existence of a new (generalized) tilting module
for stratified algebras with properly stratified Ringel dual. We show that this
new tilting module has a lot of interesting properties, for instance, its
projective dimension equals the projectively defined finitistic dimension of
the original algebra, it guarantees that the category of modules of finite
projective dimension is contravariantly finite, and, finally, it allows one to
compute the finitistic dimension of the original algebra in terms of the
projective dimension of the characteristic tilting module.
