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GaGeTe is a layered material composed of germanene and GaTe sublayers that has been recently predicted to be a basic Z2 topological semimetal. To date, only one polytype of GaGeTe is known with trigonal centrosymmetric structure (alpha phase, space group R-3m, No. 166). Here we show that as-grown samples of GaGeTe show traces of at least another polytype with hexagonal non-centrosymmetric structure (beta phase, space group P63mc, No. 186). Moreover, we suggest that another bulk hexagonal polytype (gamma phase, space group P-3m1, No. 164) could also be found near room conditions. Both alpha and beta polytypes have been identified and characterized by means of X-ray diffraction and Raman scattering measurements with the support of ab initio calculations. We provide the vibrational properties of both polytypes and show that the Raman spectrum reported for GaGeTe almost forty years ago and attributed to the alpha phase, was, in fact, that of the secondary beta phase. Additionally, we show that a Fermi resonance occurs in alpha-GaGeTe under non-resonant excitation conditions, but not under resonant excitation conditions. Theoretical calculations show that bulk beta-GaGeTe is a non-centrosymmetric weak topological semimetal with even smaller lattice thermal conductivity than centrosymmetric bulk alpha-GaGeTe. In perspective, our work paves the way for the control and engineering of GaGeTe polytypes to design and implement complex van der Waals heterostructures formed by a combination of centrosymmetric and non-centrosymmetric layers of up to three different polytypes in a single material, suitable for a number of fundamental studies and technological applications.