  Dynamical models are crucial for uncovering the internal dynamics of
galaxies, however, most of the results to date assume axisymmetry, which is not
representative for a significant fraction of massive galaxies. Here, we build
triaxial Schwarschild 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 $10^{9.5}$ to $10^{12} M_{\odot}$. We find that the
changes in internal structures within 1$R_{\rm e}$ are correlated with the
total stellar mass of the individual galaxies. The majority of the galaxies in
the sample ($73\% \pm 3\%$) are oblate, while $19\% \pm 3\%$ are mildly
triaxial and $8\% \pm 2\%$ have triaxial/prolate shape. Galaxies with $\log
M_{\star}/M_{\odot} > 10.50$ are more likely to be non-oblate. We find a mean
dark matter fraction of $f_{\rm{DM}} = 0.28 \pm 0.20$, within 1$R_{\rm e}$.
Galaxies with higher intrinsic ellipticity (flatter) are found to have more
negative velocity anisotropy $\beta_r$ (tangential anisotropy). $\beta_r$ also
shows an anti-correlation with the edge-on spin parameter \lam, so that
$\beta_r$ decreases with increasing \lam. 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 ($V/\sigma$
- $h_3$) 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.
