Using molecular dynamics simulations we investigate the structure of a system of particles interacting through a continuous core-softened interparticle potential. We found for the translational order parameter t a local maximum at a density rho(t-max) and a local minimum at rho(t-min)>rho(t-max). Between rho(t-max) and rho(t-min), the t parameter anomalously decreases upon increasing pressure. For the orientational order parameter Q(6) a maximum was observed at a density rho(t-max)<rho(Qmax)<rho(t-min). For densities between rho(Qmax) and rho(t-min), both the translational (t) and orientational (Q(6)) order parameters have anomalous behavior. We know that this system also exhibits density and diffusion anomalies. We found that the region in the pressure-temperature phase diagram of the structural anomaly englobes the region of the diffusion anomaly that is larger than the region limited by the temperature of maximum density. This cascade of anomalies (structural, dynamic, and thermodynamic) for our model has the same hierarchy as that observed for the simple point charge/extended water.