The paper presents the study of the fully mixed A (1)Sigma(+)-b (3)Pi complex of the NaRb molecule based on high-resolution sub-Doppler spectroscopy and intensity measurements, ab initio relativistic calculations of energies, transition moments and spin-orbit interactions, as well as an inverted channel-coupling approach (ICCA) deperturbation analysis. A two-laser V-type pump-probe excitation scheme was employed to obtain A<--X transition frequencies to 16 A-state vibrational levels from v=6 to v=21 with J from 8 to 23. Additionally, relative intensities in laser-induced A-->X fluorescence spectra have been recorded, including progressions with all observable transitions to the ground state vibronic levels, the latter yielding unambiguous v assignment of the A-state levels observed. All experimental rovibronic term values and all measured intensity distributions were embedded in a direct simultaneous weighted nonlinear fitting in the framework of an elaborated ICCA allowing us to obtain deperturbed relativistic diabatic potentials of the interacting A (1)Sigma(+) and b (3)Pi states. To make this possible, ab initio structure calculations of the spin-orbit singlet-triplet coupling parameter, the spin-orbit splitting of the b (3)Pi state, the transition dipole moments, and the electronic energy differences for internuclear distance 3.0-7.0 Angstrom have been performed using second order many-body multipartitioning perturbation theory. The developed ICCA is proved to be appropriate for deperturbation analysis of strongly coupled electronic states provided that accurate nonadiabatic matrix elements are known. This allows unambiguous assignment of the vibrational levels of the b (3)Pi(0) state, which is not directly observed.