In the standard situation, two standard couples face each other. The (momentarily) active lady splits the passive couple on a line slanted from the right to the left; then she walks counterclockwise around the passive lady. She splits the passive couple again, walks clockwise around the passive man, and returns to her starting position. The (momentarily) active man splits the passive couple on a line slanted from the left to the right. He then walks clockwise around the passive man, splits the passive couple again, walks counterclockwise around the passive lady, and returns to his starting position. If both actives walk simultaneously, then the lady precedes the man when they split the passive couple.
The half figure eight starts like the figure eight: The actives split the passive couple on diagonally crossing paths. (The lady has precedence at the crossing point.) They then dance around the respective contra-corner and end on the place opposite their starting point. This movement is often used to change from a proper setup to an alternating setup, or vice versa.
Quoted from my book Leitfaden / A Guide to / Contra Dance (Volume 2)
Published 2005-08-17 / Heiner Fischle, Hannover