The Mg24(t,αγ)Na23 reaction has been used to investigate the spins and decay modes of many of the levels in Na23 below an excitation energy of 6 MeV, as well as the multipole mixing ratios of the radiative transitions between these levels. γ radiation was detected in NaI(Tl) detectors located at angles between 0 and 90°, while outgoing α particles were detected in an annular detector placed at 180° with respect to the beam direction. Two-parameter techniques were used to measure the α-γ coincidence spectra. Analysis of the resultant γ- ray angular correlations, when combined with already existing data, enabled the following unambiguous spin assignments to be made: 2.98-MeV level J=32, 3.68-MeV level J=32, 3.91-MeV level J=52. Limitations on the possible spins of the 4.78-, 5.38-, and 5.74-MeV levels were obtained. The results are discussed in terms of the Nilsson model for Na23.