Molecular motors embedded within collections of actin and microtubule filaments underlie the dynamics of cytoskeletal assemblies. Understanding the physics of such motor-filament materials is critical to developing a physical model of the cytoskeleton and designing biomimetic active materials. Here, we demonstrate through experiments and simulations that the rigidity and connectivity of filaments in active biopolymer networks regulates the anisotropy and the length scale of the underlying deformations, yielding materials with variable contractility. We find that semiflexible filaments can be compressed and bent by motor stresses, yielding materials that undergo predominantly biaxial deformations. By contrast, rigid filament bundles slide without bending under motor stress, yielding materials that undergo predominantly uniaxial deformations. Networks dominated by biaxial deformations are robustly contractile over a wide range of connectivities, while networks dominated by uniaxial deformations can be tuned from extensile to contractile through cross-linking. These results identify physical parameters that control the forces generated within motor-filament arrays and provide insight into the self-organization and mechanics of cytoskeletal assemblies.