Production cross sections for the $^{35,37}\mathrm{Cl}(n,{n}^{\ensuremath{'}}\ensuremath{\gamma})$ reactions have been measured at incident neutron energies of 2.54, 3.09, and 4.03 MeV. Angular distributions of the gamma rays have been measured at 3.05 and 4.23 MeV. Gamma rays were detected with NaI(Tl) and Ge(Li) detectors using pulsed-beam techniques to eliminate backgrounds. Cylindrical and spherical samples of C${\mathrm{Cl}}_{4}$ and ${\mathrm{C}}_{2}$${\mathrm{Cl}}_{6}$ were used as well as $^{37}\mathrm{Cl}$-enriched NaCl samples. Interpretation of the angular distributions and production cross sections permitted spin assignments of \textonehalf{}, $\frac{5}{2}$, and $\frac{7}{2}$ to the first three excited levels of $^{35}\mathrm{Cl}$, and \textonehalf{} and $\frac{5}{2}$ to the first and fourth excited levels of $^{37}\mathrm{Cl}$. Excitation energies of 1.726\ifmmode\pm\else\textpm\fi{}0.004 MeV and 3.71\ifmmode\pm\else\textpm\fi{}0.02 MeV have been measured for these two levels of $^{37}\mathrm{Cl}$.