Complementary color schemes use hues directly opposite each other on the color wheel. In this silk scarf, the basic colors are derived from the hues - red and green.
Contrasting colors tend to be sharp and vibrant, especially when they are used at full saturation. However I think the complementary color scheme works well in this scarf because each dominant hue is supported with a corresponding tone for balance. Also, white is always an excellent color choice for creating a harmonious color palette.
Because of the bold colors in this scheme, I would play it safe and pair the scarf with hues from the complementary color scheme.