In polycrystals, the discontinuity of lattice rotation occurring across symmetric tilt boundaries is accommodated by the periodic arrangement of atoms in structural units. A crossover between this atomistic description and a continuous representation of tilt boundaries is carried out by designing periodic arrays of appropriately chosen smooth disclination dipoles. A comprehensive description of the boundary structure in terms of elastic strain, curvature and energy fields is then derived from a continuous theory of dislocation and disclination density fields, by allowing the initial distributions to relax in their own stress/couple stress fields. The resulting fields are obtained at nanoscale from finite element approximations of the theory. They compare remarkably well with predictions from molecular statics and experimental data. Beyond this description of grain boundaries as continua, the theory naturally provides a basis for coarse-grained spatio-temporal continuous descriptions of polycrystals.