Mmm, how many steps does it take to go from one side to the other ?
I think the following may work:
Let's say it takes 2000 steps to go from one side to the other, then you could have two layers, one with a duck facing/going to the right and the other facing/going to the left and then set the transparency as follows:
Right heading layer: $((#ZSC#/2000)%2)<=1?100:0$
Left heading layer: $((#ZSC#/2000%2)<=1?0:100$
Effectively this is determining the number of trips across the face. If it is odd then show the right. If it is even then show left.