We investigate the atmospheric dynamics of terrestrial planets in synchronous rotation within the habitable zone of low-mass stars using the Community Atmosphere Model. The surface temperature contrast between the day and night hemispheres decreases with an increase in incident stellar flux, which is opposite the trend seen in gas giants. We define three dynamical regimes in terms of the equatorial Rossby deformation radius and the Rhines length. The slow rotation regime has a mean zonal circulation that spans from the day to the night sides, which occurs for planets around stars with effective temperatures of 3300-4500 K (rotation period days), with both the Rossby deformation radius and the Rhines length exceeding the planetary radius. Rapid rotators have a mean zonal circulation that partially spans a hemisphere and with banded cloud formation beneath the substellar point, which occurs for planets orbiting stars with effective temperatures of less than 3000 K (rotation period days), with the Rossby deformation radius less than the planetary radius. In between is the Rhines rotation regime, which retains a thermally direct circulation from the day side to the night side but also features midlatitude turbulence-driven zonal jets. Rhines rotators occur for planets around stars in the range of 3000-3300 K (rotation period ∼5-20 days), where the Rhines length is greater than the planetary radius but the Rossby deformation radius is less than the planetary radius. The dynamical state can be observationally inferred from a comparison of the morphologies of the thermal emission phase curves of synchronously rotating planets.