Sexual reproduction arose somewhere around 600 million to a billion years ago as a change from cloning reproduction. By definition, in species that reproduce sexually, a single individual member of the species does not have sufficient gonadal material to reproduce by itself, but must mate with another individual of the other sexual type. Sexual reproduction means that an individual does not have sufficient sexual power in itself, and there must be another sex with which it must mate in order to reproduce.
Sexual reproduction involves an original separation of powers as well as an original division of labor. In this sense, sexes are complementary, in that both sexes are needed to complete reproduction.
( Christopher Caudwell points out that the "individual" really only originates with sexual reproduction, because in cloning no unique , new individual is produced but a copy of the parent. Sexual reproduction is a profound revolution in sociality as well).
The above gives a basis for defining a third, fourth, or x-th sex. A third sex strictly speaking would be if for a certain species three individuals of three different sexual types were NECESSARY in the mating in order to produce an offspring. Any two or one would be insufficient without the third. The three would be complementary sexes as all three are needed to complete reproduction.
CB