In this article, we study the characteristics of steady autorotation of a tethered autogyro. The phenomenon of autorotation refers to the natural spinning of a rotor in a wind field. We explore the viability of tethered autogyros as unmanned aerial vehicles (UAVs) for long-duration and energy efficient hovering applications, such as in monitoring or surveillance. The tether provides mooring and can be used to power the rotor and to transmit wind power to the ground when suitable. This is a novel application of autorotation. It requires a generalized formulation and modeling of autorotation, beyond what is reported in the literature. We adopt a model-based approach where the blade element momentum (BEM) method and catenary mechanics are used to model the aerodynamics and the tether, respectively. The resulting model is highly nonlinear and numerical methods are proposed to solve for the equilibria. The model is validated against existing simulation and experimental results in the literature. It is extended to incorporate new features that are pertinent to our application, such as low rotor speeds, regenerative torque for power generation, combining catenary mechanics with aerodynamics, and varying atmospheric conditions with altitude. We characterize the autorotational equilibria over a range of operating conditions involving multiple independent variables. The analysis reveals an optimal operating range of the tip speed ratio of the autogyro under equilibrium. It also indicates the possibility of power generation in large autogyros stationed at high altitudes.