  This paper is a step towards the complete topological classification of
{\Omega}-stable diffeomorphisms on an orientable closed surface, aiming to give
necessary and sufficient conditions for two such diffeomorphisms to be
topologically conjugate without assuming that the diffeomorphisms are
necessarily close to each other. In this paper we will establish such a
classification within a certain class {\Psi} of {\Omega}-stable diffeomorphisms
de- fined below. To determine whether two diffeomorphisms from this class
{\Psi} are topologically conjugate, we give (i) an algebraic description of the
dynamics on their non-trivial basic sets, (ii) a geometric description of how
invariant manifolds intersect, and (iii) define numerical invariants, called
moduli, associated to orbits of tangency of stable and unsta- ble manifolds of
saddle periodic orbits. This description determines the scheme of a
diffeomorphism, and we will show that two diffeomorphisms from {\Psi} are
topologically conjugate if and only if their schemes agree.
