For any $C_2$-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from $RO(C_2)$-graded cohomology to classical cohomology. In this talk, I will compare the $RO(C_2)$-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.