  We present a detailed analysis of the behavior of the triaxial Schwarzschild
orbit superposition method near the axisymmetric limit. Orbit superposition
modeling is the primary method used to determine dynamical masses of
supermassive black holes ($M_\mathrm{BH}$) in nearby galaxies; however, prior
studies have reported conflicting results when comparing the outcome from
axisymmetric orbit codes with that from a triaxial orbit code in the
axisymmetric limit. We show that in order to achieve (oblate) axisymmetry in a
triaxial code, care needs to be taken to axisymmetrize the short-axis tube
orbits and to exclude both the long-axis tube and box orbits from the orbit
library. Using up to 12 Gauss-Hermite moments of the line-of-sight velocity
distributions as constraints, we demonstrate the effects of orbit types on the
best-fit $M_\mathrm{BH}$ in orbit modeling of the massive elliptical galaxy NGC
1453 reported in Liepold et al. 2020. In addition, we verify the efficacy of
our updated code on a mock galaxy dataset. We identify a subset of slowly
precessing quasi-planar orbits for which the typical integration times can be
insufficient to fully capture the equilibrium orbital behavior in both
axisymmetric and triaxial systems with central black holes. Further
investigation is needed for a more reliable treatment of these orbits.
