Magnetic flux penetrates isotropic type II superconductors in flux-quantized vortices, which arrange themselves into a lattice structure that is independent of the direction of the applied field. In extremely anisotropic high-transition-temperature (high-Tc) superconductors, a lattice of stacks of circular 'pancake' vortices forms when a magnetic field is applied perpendicular to the copper oxide layers, while an orthogonal elongated lattice of elliptical Josephson vortices forms when the applied field is parallel to the layers. Here we report that when a tilted magnetic field is applied to single crystals of Bi2Sr2CaCu2O8+δ, these lattices can interact to form a new state of vortex matter in which all stacks of pancake vortices intersect the Josephson vortices. The sublattice of Josephson vortices can therefore be used to manipulate the sublattice of pancake vortices. This result explains the suppression of irreversible magnetization by in-plane fields as seen in Bi2Sr2CaCu2O8+δ crystals, a hitherto mysterious observation. The ability to manipulate sublattices could be important for flux-logic devices, where a 'bit' might be represented by a pancake vortex stack, and the problem of vortex positioning is overcome through sublattice interactions. This also enables the development of flux transducers and amplifiers, considerably broadening the scope for applications of anisotropic high-Tc superconductors.