Superconducting materials with noncentrosymmetric lattices lacking space inversion symmetry exhibit a variety of interesting parity-breaking phenomena, including the magnetoelectric effect, spin-polarized currents, helical states, and the unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing noncentrosymmetric superconductors with O point group symmetry, that vortices can exhibit an inversion of the magnetic field at a certain distance from the vortex core. In stark contrast to conventional superconducting vortices, the magnetic-field reversal in the parity-broken superconductor leads to nonmonotonic intervortex forces, and, as a consequence, to the exotic properties of the vortex matter such as the formation of vortex bound states, vortex clusters, and the appearance of metastable vortex/antivortex bound states.