Living cells move thanks to assemblies of actin filaments and myosin motors that range from very organized striated muscle tissue to disordered intracellular bundles. The mechanisms powering these disordered structures are debated, and all models studied so far predict that they are contractile. We reexamine this prediction through a theoretical treatment of the interplay of three well-characterized internal dynamical processes in actomyosin bundles: actin treadmilling, the attachement-detachment dynamics of myosin and that of crosslinking proteins. We show that these processes enable an extensive control of the bundle's active mechanics, including reversals of the filaments' apparent velocities and the possibility of generating extension instead of contraction. These effects offer a new perspective on well-studied in vivo systems, as well as a robust criterion to experimentally elucidate the underpinnings of actomyosin activity.