Hankel and Toeplitz X-rays of permutations

After considerable discussion intended to elucidate the connections between permutation matrices and their Hankel and Toeplitz X-rays, tournaments, transversals of partial Latin squares, and Skolem sequences, we prove several theorems concerning the existence of permutation matrices whose Hankel and Toeplitz X-rays have special properties such as being binary, palindromic, skew-palindromic, and equal. We give several methods of construction and many problems for further investigation. We also provide some numerical data obtained by computer calculation.