Twisted toroidal ribbons such as the one-sided Möbius strip have inspired scientists, engineers and artists for many centuries. A physical Möbius strip exhibits interesting mechanical properties deriving from a tendency to redistribute the torsional strain away from the twist region. This leads to the interesting possibility of building topological actuators with continuous deformations. Here we report on a series of corresponding bi-layered stripe actuators using a photothermally responsive liquid crystal elastomer as the fundamental polymeric material. Employing a special procedure, even Möbius strips with an odd number of twists can be fabricated exhibiting a seamless homeotropic and homogeneous morphology. Imposing a suitable contraction gradient under near-infrared light irradiation, these ribbons can realize continuous anticlockwise/clockwise in-situ rotation. Our work could pave the way for developing actuators and shape morphing materials that need not rely on switching between distinct states.