The ease of a metal to deform plastically in selected crystallographic planes depends on the core structure of its dislocations. As the latter is controlled by electronic interactions, metals with the same valence electron configuration usually exhibit a similar plastic behaviour. For this reason, titanium and zirconium, two transition metals of technological importance from the same column of the periodic table, have so far been assumed to deform in a similar fashion. However, we show here, using in situ transmission electron microscopy straining experiments, that plasticity proceeds very differently in these two metals, being intermittent in Ti and continuous in Zr. This observation is rationalized using first-principles calculations, which reveal that, in both metals, dislocations may adopt the same set of different cores that are either glissile or sessile. An inversion of stability of these cores between Zr and Ti is shown to be at the origin of the profoundly different plastic behaviours.