We characterized the structural and dynamical properties of the quasi-liquid layer (QLL) at the surface of ice by molecular dynamics simulations with a thermodynamically consistent water model. Our simulations show that for three low-index ice surfaces, only the outermost molecular layer presents short-range and midrange disorder and is diffusive. The onset temperature for normal diffusion is much higher than the glass temperature of supercooled water, although the diffusivity of the QLL is higher than that of bulk water at the corresponding temperature. The underlying subsurface layers impose an ordered template, which produces a regular patterning of the ice/water interface at any temperature and is responsible for the major differences between QLL and bulk water, especially for what concerns the dynamics and the midrange structure of the hydrogen-bonded network. Our work highlights the need for a holistic approach to the characterization of QLL, as a single experimental technique may probe only one specific feature, missing part of the complexity of this fascinating system.