We extend the theory of rotation vectors tohomeomorphisms of the two-dimensional torus that are homotopic to a Dehntwist. We define a one-dimensional rotation number and recreate thetheory of the homotopic case to the identity case. We prove that if such amap is area preserving and has mean rotation number zero, then it musthave a fixed point. We prove that the rotation set is a compactinterval, and that if the rotation interval contains two distinctnumbers, then for any rational number in the rotation set there exists aperiodic point with that rotation number. Finally, we prove that anyinterval with rational endpoints can be realized as therotation set of a map homotopic to a Dehn twist.