Eukaryotic nucleosomes consists of an (H3-H4)2 tetramer and two H2A-H2B dimers, around which 147 bp of DNA are wrapped in 1.7 left-handed helical turns. During chromatin assembly, the (H3-H4)2 tetramer binds first, forming a tetrasome that likely constitutes an important intermediate during ongoing transcription. We recently showed that (H3-H4)2 tetrasomes spontaneously switch between a left- and right-handed wrapped state of the DNA, a phenomenon that may serve to buffer changes in DNA torque induced by RNA polymerase in transcription. Within nucleosomes of actively transcribed genes, however, canonical H3 is progressively replaced by its variant H3.3. Consequently, one may ask if and how the DNA chirality dynamics of tetrasomes is altered by H3.3. Recent findings that H3.3-containing nucleosomes result in less stable and less condensed chromatin further underline the need to study the microscopic underpinnings of H3.3-containing tetrasomes and nucleosomes. Here we report real-time single-molecule studies of (H3.3-H4)2 tetrasome dynamics using Freely Orbiting Magnetic Tweezers and Electromagnetic Torque Tweezers. We find that the assembly of H3.3-containing tetrasomes and nucleosomes by the histone chaperone Nucleosome Assembly Protein 1 (NAP1) occurs in an identical manner to that of H3-containing tetrasomes and nucleosomes. Likewise, the flipping behavior of DNA handedness in tetrasomes is not impacted by the presence of H3.3. We also examine the effect of free NAP1, H3.3, and H4 in solution on flipping behavior and conclude that the probability for a tetrasome to occupy the left-handed state is only slightly enhanced by the presence of free protein. These data demonstrate that the incorporation of H3.3 does not alter the structural dynamics of tetrasomes, and hence that the preferred incorporation of this histone variant in transcriptionally active regions does not result from its enhanced ability to accommodate torsional stress, but rather may be linked to specific chaperone or remodeler requirements or communication with the nuclear environment.