We measure the values of individual ${\chi }^{\left(2\right)}$ tensor components in a birefringent periodically poled silica fiber through spectrally separated type I and type II second-harmonic generation. We demonstrate that the ${\chi }^{\left(2\right)}$ tensor symmetry is consistent with that of ${\chi }^{\left(3\right)}$ in silica and thereby provide experimental evidence that this ${\chi }^{\left(2\right)}$ originates from a ${\chi }^{\left(3\right)}$ process.