The surface twist, STw, is a measure of the component of duplex twist associated with the trajectory of the DNA axis wrapped on a protein surface. We calculate STw for various surfaces of revolution, including the cylinder and truncated paraboloid, ellipsoid, and hyperboloid of revolution. We show that the sign of STw cannot be stated unequivocally from knowledge of the chirality of the wrapping but depends also upon the nature of the wrapping (protein) surface. We define and discuss three geometric classes. Class (1) includes the cylinder, certain types of convex paraboloids, all concave paraboloids, the prolate ellipsoid, and the hyperboloid; here STw > 0 for right-handed wrapping and STw < 0 for left-handed wrapping. Class (2) is the sphere, for which STw = 0 for both types of handedness. Case (3) includes oblate ellipsoids and some regions of convex paraboloids; here STw < 0 for right handed wrapping and STw > 0 for left-handed wrapping.