Thanks to all who replied privately. It it the Clifford torus again. Can be seen for instance by considering the map
(z1,z2)->(z1 z2, z1 bar(z2))
where z1,z2 are complex numbers with |z1|=|z2|=1, and bar denotes complex conjugate. This maps Clifford torus onto Clifford torus (surjection) and is constant on the orbits of Z_2 acting by (z1,z2)->(-z1,-z2).
Of course the result can be obtained by indirect, more refined mathematical methods (Euler characteristic etc.)