Micrometer sized alkane-in-water emulsion drops, stabilized by appropriate long-chain surfactants, spontaneously break symmetry upon cooling and transform consecutively into series of regular shapes (Denkov et al., Nature 2015, 528, 392). Two mechanisms were proposed to explain this phenomenon of drop "self-shaping". One of these mechanisms assumes that thin layers of plastic rotator phase form at the drop surface around the freezing temperature of the oil. This mechanism has been supported by several indirect experimental findings but direct structural characterization has not been reported so far. We combine small- and wide-angle X-ray scattering (SAXS/WAXS) with optical microscopy and DSC measurements of self-shaping drops in emulsions. In the emulsions exhibiting drop self-shaping, the scattering spectra reveal the formation of intermediate, metastable rotator phases in the alkane drops before their crystallization. In addition, shells of rotator phase were observed to form in hexadecane drops, stabilized by C16EO10 surfactant. This rotator phase melts at ca. 16.6 °C which is significantly lower than the melting temperature of crystalline hexadecane, 18 °C. The scattering results are in a very good agreement with the complementary optical observations and DSC measurements.