In which phase of meiosis would certain gene segments of the homologous pairs of chromosomes "cross over" and exchange genetic information?

Amale bird-of-paradise uses a dance to attract mates in which it flaps its tail feathers on the ground and jumps around a potential female mate. a different male bird-of-paradise does a similar dance but it jumps around the female in the opposite direction. the female bird is only attracted to one style of dance, in one direction.