Equivalence classes and relations

Consider the equivalence relation defined on the unit disk $D^2={(x,y) in mathbb R^2: x^2 + y^2 le 1}$ by

$$(x_1,y_1)~ (x_2,y_2)Leftrightarrowtext{ either }(x_1,y1)=(x_2,y_2)text{ or }x_1^2 +y_1^2=x_2^2+y_2^2=1, x_1=x_2, y_1=-y_2.$$

Draw a picture showing some typical equivalence classes.