In this study we looked at spin resolved cyclotron resonance in InSb quantum wells. InSb is an attractive material for this type of work due to its small electron effective mass (0.014 mo), high intrinsic electron mobility, large electron g-factor (-51), and non-parabolic conduction band. The energies of the spin resolved transitions agreed with calculated values, even though some discrepancies in the intensities were evident. We used this spin-resolved cyclotron resonance to observe and study four subband Landau level avoided level crossings. These observed avoided level crossings are associated with the lowest Landau levels of the ground and first excited subbands of the InSb quantum wells. It was found that the energy gaps associated with anti-crossings of the same spin Landau levels depend on the tilt of the sample with respect to the magnetic field direction. This result is consistent with earlier, non spin-resolved, studies of Landau level coupling in GaAs quantum wells. We also observed anti-crossings associated with Landau Levels of opposite spin, with energy gaps that are essentially independent of sample tilt angle. Opposite spin anti-crossings are not expected from the theory explaining the GaAs experiments. Although spin-orbit effects are strong in InSb, the origins of the opposite spin anti-crossings are not definitively identified.