The Knight of Words

Question

You need to turn the following 6x6 square into a word square (i.e. so that the 1st row and 1st column both contain the same six-letter word, as do the 2nd row and 2nd column, etc.) by swapping each letter with another letter a knight’s move away from it.

T O A Y S M
S H L T T H
R E A M S E
R S E E M R
R S R Y S T
E I N E M I

How can you do it, and what are the six resulting words?

_{This puzzle is originally due to Dudeney; I found it here.}

Answer 1

The word square is

L A S H E S
A R T E R Y
S T O R M S
H E R M I T
E R M I N E
S Y S T E M

The strategy of solving it goes like this:

I identified the unique letters (L, O and N) and made the swaps to bring them on the top left to bottom right diagonal. These swaps put A and S in their positions and then looking for other swaps to bring the other A and S, mirrored across the diagonal, into position revealed more letters. When stuck, identifying a pair of letters that appears only once also helps. I identified the I pair in the bottom row and Y pair in the fourth column and worked to get them into positions such that they mirror across the diagonal. This also revealed the correct position for some other letters, and using these two techniques, all the letters slowly but surely fell into place.

Another useful technique pointed out by rand al'thor in comments:

Since the same letters must appear in symmetric positions about the diagonal, letters appearing in pairs cannot occupy the diagonal. This means, for instance that the I in the 2nd column can't move into the 1st column, and the Ts in the 2nd row can't move into the 1st row.