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After a suitable gauge fixing, the local gravitational degrees of freedom of theGowdy S1 × S2 and S3 cosmologies are encoded in an axisymmetric field onthe sphere S2. Recently, it has been shown that a standard field parametrizationof these reduced models admits no Fock quantization with a unitary dynamics.This lack of unitarity is surpassed by a convenient redefinition of the fieldand the choice of an adequate complex structure. The result is a Fockquantization where both the dynamics and the SO(3)-symmetries of the fieldequations are unitarily implemented. The present work proves that this Fockrepresentation is in fact unique inasmuch as, up to equivalence, there exists noother possible choice of SO(3)-invariant complex structure leading to a unitaryimplementation of the time evolution.