For rotational rheometers, surface tension results in a torque that should not occur in an ideal, rotationally symmetric geometry. This paper identifies and explains the effect, which is due to surface tension and contact line traction forces, and not surface rheology at the liquid-air interface. For water, we show that this torque can be more than two orders of magnitude larger than the viscous torque. In steady shear flow, the effect appears as a constant torque independent of rate, which would appear inaccurately as apparent shear thinning of water. In oscillatory tests, this may appear inaccurately as an elastic modulus. This surface tension torque is sensitive to wetting conditions and contact line asymmetries and cannot be deterministically corrected in experimental measurements. It therefore raises the lower bound of the instrument low-torque limit. The surface tension torque is reduced by maximizing rotational symmetry of the contact line, minimizing evaporation and the migration of the contact line, reducing the radial location of the contact line, and lowering the surface tension. Identifying and eliminating the surface tension torque is critical for low viscosities, intrinsic viscosities, soft materials, subdominant viscoelastic components, small gaps, and any circumstance where the low-torque limit is experimentally important. [ABSTRACT FROM AUTHOR]