  The two tableaux assigned by the Robinson--Schensted correspondence are equal
if and only if the input permutation is an involution, so the RS algorithm
restricts to a bijection between involutions in the symmetric group and
standard tableaux. Beissinger found a concise way of formulating this
restricted map, which involves adding an extra cell at the end of a row after a
Schensted insertion process. We show that by changing this algorithm slightly
to add cells at the end of columns rather than rows, one obtains a different
bijection from involutions to standard tableaux. Both maps have an interesting
connection to representation theory. Specifically, our insertion algorithms
classify the molecules (and conjecturally the cells) in the pair of $W$-graphs
associated to the unique equivalence class of perfect models for a generic
symmetric group.
