TITLE: Divide a square into rectangles where each occurs once in each orientation
QUESTION [12 upvotes]: A $26 \times 26$ square divides into different rectangles so that each occurs exactly twice in different orientations.

I've also found a solution for the $10 \times 10$ square, but no others.  Are there any other squares that can be divided into a finite number of rectangles so that each occurs exactly twice in different orientations?

REPLY [2 votes]: Let us consider the following tiling of $8 \times 8$ square:
11111222
11111222
11111222
55557222
64497222
64493333
64493333
64400088

Each digit indicates number of rectangle the cell belongs to.
From 0 to 9 the sizes are: $3 \times 1$, $3 \times 5$, $5 \times 3$, $4 \times 2$, $2 \times 4$, $4 \times 1$, $1 \times 4$, $1 \times 2$, $2 \times 1$, $1 \times 3$.