We report a measurement of two energy-weighted γ cascade angular distributions from polarized slow neutron capture on the ³⁵Cl nucleus, one parity-odd correlation proportional to s^→_n · k^→_γ and one parity-even correlation proportional to s^→_n · k^→_n × k^→_γ. A parity-violating asymmetry can appear in this reaction due to the weak nucleon-nucleon interaction, which mixes opposite parity S- and P-wave levels in the excited compound ³⁶Cl nucleus formed upon slow neutron capture. If parity-violating (PV) and parity-conserving (PC) terms both exist, the measured differential cross section can be related to them via ^dσ/_dΩ ∝ 1 + A_γ,_PVcosθ + A_γ,_PCsinθ. The PV and PC asymmetries for energy-weighted γ cascade angular distributions for polarized slow neutron capture on ³⁵Cl averaged over the neutron energies from 2.27-9.53 meV were measured to be A_γ,PV = (-23.9 ± 0.7) × 10^-6 and A_γ,PC = (0.1 ± 0.7) × 10^-6. These results are consistent with previous experimental results. Systematic errors were quantified and shown to be small compared to the statistical error. These asymmetries in the angular distributions of the γ rays emitted from the capture of polarized neutrons in ³⁵Cl were used to verify the operation and data analysis procedures for the NPDGamma experiment, which measured the parity-odd asymmetry in the angular distribution of γs from polarized slow neutron capture on protons.