A new adiabatic representation of the three-body wave function obtained in terms of the local adiabatic expansions of the Faddeev components is investigated. As a result, we construct a unique adiabatic basis describing all possible channels of the three-body system with inclusion of break-up and rearrangement ones. Moreover, we give a complete classification of the basis states and the corresponding surface functions. The implementation of simple geometrical and spectral arguments allows us to work out a global adiabatic approach to the three-body scattering problem for the class of pair short-range potentials. Therein we formulate the three-body scattering problem by consistently reducing it to the multichannel radial and parametric quasiangular ones, including the correct boundary conditions with account of break-up and rearrangement processes. We also state the inverse three-body problem. 25 refs.