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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.