Now cross-posted to MO because of no answers here.
The following Latin square
$$\begin{bmatrix} 1&2&3&4&5&6&7&8\\ 2&1&4&5&6&7&8&3\\ 3&4&1&6&2&8&5&7\\ 4&3&2&8&7&1&6&5\\ 5&6&7&1&8&4&3&2\\ 6&5&8&7&3&2&4&1\\ 7&8&5&2&4&3&1&6\\ 8&7&6&3&1&5&2&4 \end{bmatrix}$$
has the property that for all pairs of two different rows $a$ and $b$, the permutations $ab^{-1}$ have the same cycle type (one 2-cycle and one 6-cycle).
What is known about such Latin squares? (With the property that all $ab^{-1}$ have the same cycle type, not necessarily the (2, 6) cycle type from the example above.) For example, do they have a particular structure, for which cycle types do they exist, do they have a name, etc?