We report on the extensional dynamics of spinning drops in miscible and immiscible background fluids following a rotational speed jump. Two radically different behaviors are observed. Drops in immiscible environments relax exponentially to their equilibrium shape, with a relaxation time that does not depend on the centrifugal force. We find an excellent quantitative agreement with the relaxation time predicted for quasi-spherical drops by Stone and Bush (Q. Appl. Math. 1996, 54, 551), while other models proposed in the literature fail to capture our data. By contrast, drops immersed in a miscible background fluid do not relax to a steady shape: they elongate indefinitely, their length following a power-law in very good agreement with the dynamics predicted by Lister and Stone (J. Fluid Mech. 1996, 317, 275) for inviscid drops. Our results strongly suggest that low compositional gradients in miscible fluids do not give rise to an effective interfacial tension measurable by spinning drop tensiometry.