The structure of the nematic (cholesteric) drops that form at the clearing temperature of a mixture of the bent-core molecule CB7CB and the rodlike molecule 8CB doped with a surfactant is optically determined. Using experimental observations and numerical simulations, it is demonstrated that the director field inside these drops is not escaped concentric, as previously proposed, but twisted bipolar. The Lehmann rotation of these drops in the presence of a temperature gradient is described. Their rotation velocity is shown to be proportional to the temperature gradient and to the surface twist angle of the director field and inversely proportional to the drop radius, thus revealing a fundamental scaling law for the Lehmann effect of nematic and cholesteric twisted-bipolar droplets.