Local structures play a crucial role in glass formation and properties. In addition to topological short-range order, the geometric property of site symmetry is another important but less known characteristic of local structures. It is shown that the observed sharp increase of glass forming ability of Ce70-xAl10Cu20Cox upon Co addition is correlated with a dramatic increase of Al site symmetry, as reflected by decreasing quadrupole frequency measured by 27Al NMR. The result is consistent with the structure model of Al-centered icosahedral clusters as the predominant structural building blocks.