I have in front of me the official Rules of the Game for Kamisado. They make me wonder which player takes the opening move in a new round after filling from the left or the right.
The section "Filling from the Left or Rights" explains that the towers "can be set up in one of two ways, determined by the player who won the previous round. This player will be the Defender in the new round". So this would suggest that the player who lost the previous round takes the first turn in the next round. That's also what seems to be implemented e.g. in the online version at Yucata.
However the description of Figure 7(h) states that the "diagram shows the possible moves open to Alain with his purple Sumo dragon tower, should he decide to make his opening move with this tower". Alain has just gained a Sumo tower by winning a round, so this would suggest that the player who won the previous round takes the first turn in the next round. Only if Alain were to begin could he chose which tower to make his opening move with.
I suspect that 1. is the right answer but am not yet sure. Who knows better and can explain the apparently odd description for Figure 7(h)?