The transport properties of magnetic tunnel junctions (MTJs) are very sensitive to interface modifications. In this work we investigate both experimentally and theoretically the effect of asymmetric barrier modifications on the bias dependence of tunneling magnetoresistance (TMR) in single crystal Fe/MgO-based MTJs with (i) one crystalline and one rough interface, and (ii) with a monolayer of O deposited at the crystalline interface. In both cases we observe an asymmetric bias dependence of TMR and a reversal of its sign at large bias. We propose a general model to explain the bias dependence in these and similar systems reported earlier. The model predicts the existence of two distinct TMR regimes: (i) a tunneling regime when the interface is modified with layers of a different insulator, and (ii) a resonant regime when thin metallic layers are inserted at the interface. We demonstrate that in the tunneling regime, negative TMR is due to the high voltage which overcomes the exchange splitting in the electrodes, while the asymmetric bias dependence of TMR is due to the interface transmission probabilities. In the resonant regime, inversion of TMR could happen at zero voltage depending on the alignment of the resonance levels with the Fermi surfaces of the electrodes. Moreover, the model predicts a regime in which TMR has different signs at positive and negative bias, suggesting possibilities of combining memory with logic functions.