Recently, as part of some joint research, Tom Roby was led to a curious operation on strings of L's and R's which he calls "bounce-reading": We start by reading the string at the left. When the symbol we have just read is an L, the next symbol we read is the leftmost unread symbol; but when the symbol we have just read is an R, the next symbol we read is the rightmost unread symbol. For instance, if our string of L's and R's is LLRRLRLR, we read the positions in the order 1,2,3,8,7,4,6,5, obtaining the bounce-reading LLRRLRRL.
It may be more natural to think of this as an action on circular words; when we read a letter, we delete it, close up the circle, and move either to the left or the right of the deletion-point.
Has this operation been studied before?