In European medieval and renaissance love poetry, the handkerchief is typically a symbol for a woman's romantic favor
There was a particular ritual in which a lady would drop her handkerchief for a knight to pick up and keep as a token of her regard
The handkerchief that Othello gives Desdemona is, similarly, a love-token and a symbol of their love
The handkerchief, which originally belonged to an Egyptian sorcerer, also comes to symbolize the illusions that Iago is "casting" through his plotting