We report equilibrium and nonequilibrium molecular dynamics (MD) simulations of two nucleosome core particles (NCPs) stacked with their dyad axes oriented in parallel or antiparallel fashion. From the equilibrium trajectories, we determine the bridging behavior of different histone tails and observe that different sets of histone tails play important roles in the two orientations in stabilizing the NCP stack. While the H4 and H2A tails play important intermediary roles in the parallel stack, the H3 and H2B tails are important in the antiparallel stack. We use steered MD simulations to unstack the two NCPs and find a stark difference in their unstacking pathways. While the average rupture force was found to be higher for the parallel stack, the work done for complete unstacking was similar for both orientations. We use Jarzynski equality to determine the PMF profiles along the unstacking pathway, relate our findings to the behavior of NCP mesophases, and derive insights into the enigmatic nucleosomal organization in the chromatin fiber.