Methinks Elvis is in wind-up mood tonight. (Any chicken that lies on either side is probably dead. They sleep in a squatting position).
The factory that sends you the chicken halves almost certainly provides legs, breasts and thighs to other people. The guy with the cleaver (or the cutting machine if it's a big operation) probably has the bins for legs, breasts and thighs on one side of his work area, and the bin for half chickens on the other. So he starts by cutting the chicken in half and throwing a half-chicken in the relevant bin. Then he slices up the remaining half and put the portions in the relevant bins.
Chris
PS: (For Shanx): All of the best people are left-handed. Like me, both my parents, all of my grandparents, all of my uncles and aunts (even by marriage!) and all of my cousins ;-)