In eukaryotes, DNA is packaged within nucleosomes. The DNA of each nucleosome is typically centered around an octameric histone protein core: one central tetramer plus two separate dimers. Studying the assembly mechanisms of histones is essential for understanding the dynamics of entire nucleosomes and higher-order DNA packaging. Here, we investigate canonical histone assembly and that of the centromere-specific histone variant, centromere protein A (CENP-A), using molecular dynamics simulations. We quantitatively characterize their thermodynamical and dynamical features, showing that two H3/H4 dimers form a structurally floppy, weakly bound complex, the latter exhibiting large instability around the central interface manifested via a swiveling motion of two halves. This finding is consistent with the recently observed DNA handedness flipping of the tetrasome. In contrast, the variant CENP-A encodes distinctive stability to its tetramer with a rigid but twisted interface compared to the crystal structure, implying diverse structural possibilities of the histone variant. Interestingly, the observed tetramer dynamics alter significantly and appear to reach a new balance when H2A/H2B dimers are present. Furthermore, we found that the preferred structure for the (CENP-A/H4) tetramer is incongruent with the octameric structure, explaining many of the unusual dynamical behaviors of the CENP-A nucleosome. In all, these data reveal key mechanistic insights and structural details for the assembly of canonical and variant histone tetramers and octamers, providing theoretical quantifications and physical interpretations for longstanding and recent experimental observations. Based on these findings, we propose different chaperone-assisted binding and nucleosome assembly mechanisms for the canonical and CENP-A histone oligomers.