Each time a cell divides, the microtubule cytoskeleton self-organizes into the metaphase spindle: an ellipsoidal steady-state structure that holds its stereotyped geometry despite microtubule turnover and internal stresses [1-6]. Regulation of microtubule dynamics, motor proteins, microtubule crosslinking, and chromatid cohesion can modulate spindle size and shape, and yet modulated spindles reach and hold a new steady state [7-11]. Here, we ask what maintains any spindle steady-state geometry. We report that clustering of microtubule ends by dynein and NuMA is essential for mammalian spindles to hold a steady-state shape. After dynein or NuMA deletion, the mitotic microtubule network is "turbulent"; microtubule bundles extend and bend against the cell cortex, constantly remodeling network shape. We find that spindle turbulence is driven by the homotetrameric kinesin-5 Eg5, and that acute Eg5 inhibition in turbulent spindles recovers spindle geometry and stability. Inspired by in vitro work on active turbulent gels of microtubules and kinesin [12, 13], we explore the kinematics of this in vivo turbulent network. We find that turbulent spindles display decreased nematic order and that motile asters distort the nematic director field. Finally, we see that turbulent spindles can drive both flow of cytoplasmic organelles and whole-cell movement-analogous to the autonomous motility displayed by droplet-encapsulated turbulent gels . Thus, end-clustering by dynein and NuMA is required for mammalian spindles to reach a steady-state geometry, and in their absence Eg5 powers a turbulent microtubule network inside mitotic cells.