We investigate the spatiotemporal structure of the biphoton correlation in type-II parametric down-conversion (PDC). As in type-I PDC [Phys. Rev. Lett. 102, 223601 (2009)], we find that the correlation is nonfactorizable in space and time. Differently from type I, the type-II correlation in the spontaneous regime displays an asymmetric V-shaped structure in any cross section including time and one transverse dimension. This asymmetry along the temporal coordinate originates from the signal-idler group velocity mismatch and tends to disappear as the parametric gain is raised. We observe a progressive transition toward a symmetric X-shaped geometry similar to that found in type I when stimulated PDC becomes dominant. We also give quantitative evaluations of the localization in space and in time of the correlation, analyze its behavior for different crystal tuning angles, and underline qualitative differences with respect to type-I PDC.