The KL right cell is the collection of permutations that have the same insertion tableau S from RSK algorithm, the left cell is those of the same recording tableau T.
KL right(left) cell are the equivalence classes of KL right(left) preorder on $\mathfrak{S}_n$.

The S of 312 is :
$$1 \;\; 2$$
$$3\;\;\;$$
The T is:$$1 \;\; 3$$
$$2\;\;\;$$