Kinesin is a homodimeric motor with two catalytic heads joined to a stalk via short neck linkers (NLs). We measured the torsional properties of single recombinant molecules by tracking the thermal angular motions of fluorescently labeled beads bound to the C terminus of the stalk. When kinesin heads were immobilized on microtubules (MTs) under varied nucleotide conditions, we observed bounded or unbounded angular diffusion, depending on whether one or both heads were attached to the MT. Free rotation implies that NLs act as swivels. From data on constrained diffusion, we conclude that the coiled-coil stalk domains are approximately 30-fold stiffer than its flexible "hinge" regions. Surprisingly, while tracking processive kinesin motion at low ATP concentrations, we observed occasional abrupt reversals in the directional orientations of the stalk. Our results impose constraints on kinesin walking models and suggest a role for rotational freedom in cargo transport.