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.