Assuming that the parametrization of the profile curve is
the corresponding twisted surface of revolution has parametrization
is the amount of twisting (
yields an ordinary surface of revolution).
The first three of the six examples in this Demonstration are particularly noteworthy:
"semicircle" has its center on the
axis and gives a corkscrew surface (twisted sphere);
"tractrix" is a Dini's surface (twisted pseudosphere) that is a well-known example of a surface with constant negative Gaussian curvature;
"zero-curvature" is the curve with
that produces a surface with constant zero Gaussian curvature.
See, for example, [1, Chapter 15] and [2, Exercise 4.46].
 A. Gray, E. Abbena and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica
, 3rd ed., Boca Raton, FL: Chapman & Hall/CRC, 2006.
 K. Tapp, Differential Geometry of Curves and Surfaces
, New York: Springer Science+Business Media, 2016.