Abstract Dynamical models are crucial for uncovering the internal dynamics of galaxies; however, most of the results to date assume axisymmetry, which is not representative of a significant fraction of massive galaxies. Here, we build triaxial Schwarzschild orbit-superposition models of galaxies taken from the SAMI Galaxy Survey, in order to reconstruct their inner orbital structure and mass distribution. The sample consists of 161 passive galaxies with total stellar masses in the range 10^9.5–10¹² M _⊙. We find that the changes in internal structures within 1R _e are correlated with the total stellar mass of the individual galaxies. The majority of the galaxies in the sample (73% ± 3%) are oblate, while 19% ± 3% are mildly triaxial and 8% ± 2% have triaxial/prolate shape. Galaxies with log M ⋆ / M ⊙ > 10.50 are more likely to be non-oblate. We find a mean dark matter fraction of f _DM = 0.28 ± 0.20, within 1R _e. Galaxies with higher intrinsic ellipticity (flatter) are found to have more negative velocity anisotropy β _r (tangential anisotropy). β _r also shows an anticorrelation with the edge-on spin parameter λ Re , EO , so that β _r decreases with increasing λ Re , EO , reflecting the contribution from disk-like orbits in flat, fast-rotating galaxies. We see evidence of an increasing fraction of hot orbits with increasing stellar mass, while warm and cold orbits show a decreasing trend. We also find that galaxies with different (V/σ – h ₃) kinematic signatures have distinct combinations of orbits. These results are in agreement with a formation scenario in which slow- and fast-rotating galaxies form through two main channels.