Microtubules and molecular motors are essential components of the cellular cytoskeleton, driving fundamental processes in vivo, including chromosome segregation and cargo transport. When reconstituted in vitro, these cytoskeletal proteins serve as energy-consuming building blocks to study the self-organization of active matter. Cytoskeletal active gels display rich emergent dynamics, including extensile chaotic flows, locally contractile asters, and bulk contraction. However, it is unclear which protein-scale interactions, if any, set the contractile or extensile nature of the active gel, or how to control the transition from one phase to another in a simple active system. Here, we explore the microscopic origin of the extensile-to-contractile transition in a reconstituted active material composed of stabilized microtubules, depletant, ATP, and clusters of kinesin-1 motors. We show that microscopic interactions between highly processive motor clusters and microtubules can trigger the end-accumulation of motor proteins, which in turn drives polarity sorting and aster formation. Combining experiments and simple scaling arguments, we demonstrate that the extensile-to-contractile transition is akin to a self-assembly process where nematic and polar aligning interactions compete. We identify a single control parameter given by the ratio of the different component concentrations that dictates the material-scale organization into either a contractile aster or an extensile bundle. Overall, this work shows that robust self-organization of cytoskeletal active materials can be controlled by precise biochemical and mechanical tuning at the microscopic level.