Data in support of research: Stratospheric aerosol injection (SAI), as a possible supplement to emission reduction, has the potential to reduce some of the risks associated with climate change. Adding aerosols to the lower stratosphere results in global cooling. However, different choices for the aerosol injection latitude(s) and season(s) have been shown to lead to significant differences in regional surface climate, introducing a design aspect to SAI. Past research has shown that there are at least three independent degrees of freedom (DOF) that can be used to simultaneously manage three different climate goals. Knowing how many more DOFs there are, and thus how many independent climate goals can be simultaneously managed, is essential to understanding fundamental limits of how well SAI might compensate for anthropogenic climate change, and evaluating any underlying trade-offs between different climate goals. Here we quantify the number of meaningfully-independent DOFs of the SAI design space. This number of meaningfully-independent DOFs depends on both the amount of cooling and the climate variables used for quantifying the changes in surface climate. At low levels of global cooling, only a small set of injection choices yield detectably different surface climate responses. For a cooling level of 1-1.5℃, we find that there are likely between 6 and 8 meaningfully-independent DOFs. This narrows down the range of available DOF and also reveals new opportunities for exploring alternate SAI designs with different distributions of climate impacts.