We propose the first correct special-purpose quantum circuits for preparation of Bell-diagonal states (BDS), and implement them on the IBM Quantum computer, characterizing and testing complex aspects of their quantum correlations in the full parameter space. Among the circuits proposed, one involves only two quantum bits but requires adapted quantum tomography routines handling classical bits in parallel. The entire class of Bell-diagonal states is generated, and a number of characteristic indicators, namely entanglement of formation, CHSH non-locality, steering and discord, are experimentally evaluated over the full parameter space and compared with theory. As a by-product of this work we also find a remarkable general inequality between "quantum discord" and "asymmetric relative entropy of discord": the former never exceeds the latter. We also prove that for all BDS the two coincide.