Proteins, molecules, and macromolecular assemblies in water are surrounded by a nanometer-sized hydration layer with properties very different from bulk water. Here, we use classical molecular dynamics simulations to study the dielectric response of hydration water next to hydrophobic and hydrophilic planar surfaces. We find the interfacial dielectricabsorption of water to be strongly anisotropic: compared to bulk water, which shows a broad dielectricabsorption maximum around 15 GHz in the imaginary part of the dielectric function, the absorption for electric fields parallel to the surface is of similar strength and shows a slight redshift, while for perpendicular electric fields it is strongly attenuated and blueshifted. This anisotropy is generic for hydrophobic and hydrophilic surfaces. From our spatially resolved dielectric functions and a modified Maxwell-Garnett theory that accounts for anisotropic hydration layers around spherical particles, the dielectricabsorption of solutions of organic molecules and micelles is derived to exhibit the experimentally known attenuation in combination with a redshift. These two features are traced back to the subtle interplay of interfacial depolarization effects and the dielectricanisotropy in the hydration layer. By a detailed analysis of the individual water molecule dynamics the perpendicular blueshift is shown not to be linked to accelerated water reorientation, but rather to dielectric boundary effects. Carefully conducted angularly resolved experiments at planar aqueous interfaces will be able to resolve this dielectricanisotropy and thus to confirm the subtle connection between spectralabsorption features and the molecular water dynamics in hydration layers.