We study the dynamic behavior in a vertical-cavity surface-emitting laser subject to orthogonal optical injection through the computation of Lyapunov exponents and isospikes for a wide range of intervals in the plane of the injection parameters, i.e., the frequency detuning vs injection strength plane. Our thorough numerical experiments on this plane constitute a deep quantitative analysis of the different bifurcation scenarios leading to polarization switching (PS). First, we obtain similar results for the linearly polarized intensities for the different PS scenarios, especially when the injection strength is increased. It allows us to determine the parameter values that will be used for further analysis of the bifurcation scenarios in the parameter space. Analysis of different phase diagrams enables us to show multistability in the system and identify in the parameter planes several regions such as the predominantly chaotic lobe ones inside them are embedded some mainly regular structures such as spirals, rings, tricorns, shrimp networks, "eye(s) of chaos," and chiral and nonchiral distribution of periodicities characterized by sequences of quint points. We emphasize two routes to chaos, namely, period-doubling and quint-point-based bifurcations.