Abstract A phenomenological exploration of the low-energy behaviour of dips in some hadronic two-body processes (π p, Kp, pp scatterings and Λ production) is presented. It is found that such behaviour is in most cases unexpectedly simple, and that in several cases dips and resonances describe simple patterns as the bare Veneziano formula suggests. Phase shifts are systematically used, and prove to be of critical importance to get an understanding of the low energy dip structure. The results of the study support physical significance of the Veneziano formula and contribute to clarify its phenomenological role. The view that the formula already contains important contributions from unitarity appears to be encouraged. On the other hand, the apparent limits of applicability of the formula seem to deflate the problem of avoiding parity doublets in the theory.