Aims. We study the geometry of the circumstellar environment of the B[e] supergiant star GG Car. Methods: We present observations acquired using the IAGPOL imaging polarimeter in combination with the Eucalyptus-IFU spectrograph to obtain spectropolarimetric measurements of GG Car across Hα at two epochs. Polarization effects along the emission line are analysed using the Q-U diagram. In particular, the polarization position angle (PA) obtained using the line effect is able to constrain the symmetry axis of the disk/envelope. Results: By analysing the fluxes, GG Car shows an increase in its double-peaked Hα line emission relative to the continuum within the interval of our measurements (~43 days). The depolarization line effect around Hα is evident in the Q-U diagram for both epochs, confirming that light from the system is intrinsically polarized. A rotation of the PA along Hα is also observed, indicating a counter-clockwise rotating disk. The intrinsic PA calculated using the line effect (~85°) is consistent between our two epochs, suggesting a clearly defined symmetry axis of the disk. Based on observations obtained at the Observatório do Pico dos Dias, LNA/MCT, Itajubá, Brazil.