Packing of permutations into Latin squares

For every positive integer $n$ greater than $4$ there is a set of Latin squares of order $n$ such that every permutation of the numbers $1,\ldots,n$ appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If $n$ is greater than $4$ and not of the form $p$ or $2p$ for some prime number $p$ congruent to $3$ modulo $4$, then there always exists a Latin square of order $n$ in which the rows, columns, reverse rows and reverse columns are all distinct permutations of $1,\ldots,n$, and which constitute a permutation group of order $4n$. If $n$ is prime congruent to $1$ modulo $4$, then a set of $(n-1)/4$ mutually orthogonal Latin squares of order $n$ can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order $n(n-1)$.