We study the attractive interactions between rodlike charged polymers in solution that appear in the presence of multivalence counterions. The counterions condensed to the rods exhibit both a strong transversal polarization and a longitudinal crystalline arrangement. At short distances between the rods, the fraction of condensed counterions increases, and the majority of these occupy the region between the rods, where they minimize their repulsive interactions by arranging themselves into packing structures. The attractive interaction is strongest for multivalent counterions. Our model takes into account the hard-core volume of the condensed counterions, and their angular distribution around the rods. The hard-core constraint strongly suppresses longitudinal charge fluctuations.