Blue and green should never be seen together because the colors clash. However, the saying continues that blue and green should never be seen unless there's a color between, indicating that a small separation makes the pairing OK. This old saying has kept the two colors apart in many outfits and decorating schemes over the years, but some fashion lovers have rebelled from this idea recently by pairing the colors.
On a basic color wheel, blue and green are located right next to each other, but that doesn't mean that the colors look good together from a visual perspective. If the color wheel is broken into 12 segments, any three adjacent slices can be put together to create an analogous color scheme. In this case, blue and green can be displayed beautifully together, but they have a blue-green or turquoise slice between them, and this works to visually soothe the transition between the colors.
Blue and green are certainly not considered to be complementary because complementary colors have to be located opposite each other on a color wheel. Thus, purple is complementary to green, and orange to blue.
There are similar phrases discouraging other color combinations. One such phrase is "red and green should only be seen on an Irish queen."