At each cell division, nanometer-scale motors and microtubules give rise to the micron-scale spindle. Although the biophysical properties of many individual motors are well studied, it is not fully understood how these molecular components push, pull, and twist microtubules to build a spindle with emergent mechanics and robust function, or how these processes are subverted in disease. We first focus on the contractile motor dynein and the extensile motor Eg5. Although these motors are individually required to build the human spindle, typical bipolar spindles form when both are inhibited together, raising the question of what roles these opposing motors play. Using quantitative live imaging, we find that co-inhibiting Eg5 and dynein’s targeting factor NuMA generates spindles that attain a normal metaphase shape and undergo anaphase. However, these spindles exhibit reduced microtubule dynamics, mechanical fragility, and error-prone chromosome segregation. Thus, although these opposing motors are not required for spindle assembly, they are essential to the spindle's mechanical and functional robustness. Unexpectedly, we also find that spindles lacking NuMA/dynein and Eg5 activity display strong left-handed twist at anaphase, leading us to ask how molecular torques generated by chiral mitotic motors are balanced to regulate global spindle twist. We find that the midzone motors KIF4A and MKLP1 contribute to left-handed twist at anaphase, as does the actin cytoskeleton. Depletion of LGN or dynein, which generate force at the cell cortex, counteracts these left-handed torques. Thus, competing torques are balanced to establish the spindle’s global twist, and torques acting from the outside of the spindle balance those generated within it. Finally, we investigate how oncogenic transformation affects spindle architecture and mechanics. We find that cyclin D1 overexpression increases the incidence of spindles with extra poles, centrioles, and chromosomes, but that it also protects spindle poles from fracturing under compressive forces. We propose that cyclin D1 overexpression may be adaptive in stiff solid tumors, contributing to its prevalence in cancer by allowing continued proliferation in mechanically challenging environments. Together, this work reveals design principles by which the spindle achieves robustness, and how they are rewired in disease.