The Fermi surface (FS) of the Ag/Si(111)-3×3 reconstruction with an excess of Ag has been mapped by angle resolved photoemission spectroscopy with polarized light in a wide region of the reciprocal space and with different detection geometries. In contrast to previous results, a strong polarization dependence is observed. Applying the dipole selection rules, it is found that the surface state at the Fermi level, S1 state, has odd symmetry with respect to the mirror plane of the honeycomb-chained triangle structure, indicating that it is mainly derived from Ag 5px and 5py orbitals. This conclusion is revised in the new frame of a inequivalent-triangle structure for the Ag/Si(111)-3×3 at room temperature. Besides, strong modulations of the intensity distribution are found that deviate the Fermi surface pattern from its expected two-dimensional periodical behavior.