It has been proposed recently that the first step in the formation of both rocky and gas giant planets is dust sedimentation into a solid core inside a gas clump (giant planet embryo). The clumps are then assumed to migrate closer to the star where their metal poor envelopes are sheared away by the tidal forces or by an irradiation-driven mass loss. We consider the implications of this hypothesis for natal rotation rates of both terrestrial and gas giant planets. It is found that both types of planets may rotate near their breakup angular frequencies at birth. The direction of the spin should coincide with that of the parent disc and the star, except in cases of embryos that had close interactions or mergers with other embryos in the past. Furthermore, the large repository of specific angular momentum at birth also allows formation of close binary rocky planets inside the same embryos. We compare these predictions with rotation rates of planets in the Solar system and also question whether the Earth–Moon pair could have been formed within the same giant planet embryo.