Colloidal gels are formed by the aggregation of Brownian particles into clusters that are, in turn, part of a space-spanning percolated network. In practice, the microstructure of colloidal gels, which dictates their mechanical properties, strongly depends on the particle concentration and on the nature of their interactions. Yet another critical control parameter is the shear history experienced by the sample, which controls the size and density of the cluster population, via particle aggregation, cluster breakup, and restructuring. Here, we investigate the impact of shear history on acid-induced gels of boehmite, an aluminum oxide. We show that following a primary gelation, these gels display a dual response depending on the shear rate γ˙p used to rejuvenate their microstructure. We identify a critical shear rate γ˙c, above which boehmite gels display a gel-like viscoelastic spectrum upon flow cessation, similar to that obtained following the primary gelation. However, upon flow cessation after shear rejuvenation below γ˙c, boehmite gels display a glassylike viscoelastic spectrum together with enhanced elastic properties. Moreover, the nonlinear rheological properties of boehmite gels also differ on both sides of γ˙c: weak gels obtained after rejuvenation at γ˙p>γ˙c show a yield strain that is constant, independent of γ˙p, whereas strong gels obtained with γ˙p<γ˙c display a yield strain that significantly increases with γ˙p. Our results can be interpreted in light of the literature on shear-induced anisotropy, which accounts for the reinforced elastic properties at γ˙p<γ˙c, while we rationalize the critical shear rate γ˙c in terms of a dimensionless quantity, the Mason number, comparing the ratio of the strength of the shear flow with the interparticle bond force.