ABSTRACTMost dynamical models of galaxies to date assume axisymmetry, which is not representative of a significant fraction of massive galaxies. We have built triaxial orbit-superposition Schwarzschild models of galaxies observed by the SAMI Galaxy Survey, in order to reconstruct their inner orbital structure and mass distribution. The sample consists of 153 passive galaxies with total stellar masses in the range 109.5 to $10^{12} \, {\rm M}_{⊙ }$. We present an analysis of the internal structures and intrinsic properties of these galaxies as a function of their environment. We measure their environment using three proxies: central or satellite designation, halo mass and local 5th nearest neighbour galaxy density. We find that although these intrinsic properties correlate most strongly with stellar mass, environment does play a secondary role: at fixed stellar mass, galaxies in the densest regions are more radially anisotropic. In addition, central galaxies, and galaxies in high local densities show lower values of edge-on spin parameter proxy λRe, EO. We also find suggestions of a possible trend of the fractions of orbits with environment for lower mass galaxies (between 109.5 and $10^{11} \, {\rm M}_{⊙ }$) such that, at fixed stellar mass, galaxies in higher local densities and halo mass have higher fractions of hot orbits and lower fractions of warm orbits. Our results demonstrate that after stellar mass, environment does play a role in shaping present-day passive galaxies.