Consider Latin squares of even order that is not of form $2^x$, where every cell is involved in a $2\times 2$ sub square. Here is one such square for order 6:
0 1 2 3 4 5 1 0 3 4 5 2 2 4 0 5 1 3 3 5 1 2 0 4 4 2 5 1 3 0 5 3 4 0 2 1
Notice that rows correspond to each other in pairings. Cells in the first row make sub squares with cells in the third row, cells in the second row make sub squares with cells in the fourth, and the fifth and sixth rows make sub squares with each other. These are not the only existent subsquares, but there is a definitive correspondence between pairs of rows for subsquares.
The question- what does a latin square of order 10 with all cells involved in a $2\times 2$ subsquare, and with rows presumably paired like in the order 6 square satisfying the same conditions, look like? And once we can see that square, is there any visible relationship between such squares of even but not power-of-two order and the group tables for $\mathbb{Z}/2^n$, which also satisfy the all-cells-involved-in-two-by-two-subsquares condition? Is there any generalizable pattern for this sort of square at all?
But I suppose my main question is with regards to finding $10\times 10, 12\times 12$ etc examples- especially $10\times 10$. I have not been able to find an example using the pairing rows method- maybe I am just being disorganized.