Divide a square into rectangles where each occurs once in each orientation

Question

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?

Answer 1

Here are all rectangles with area 2 through 13, both orientations of those that are not square, in a 17x17. Not quite what you requested since I left the squares in.

Answer 2

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: $3 \times 1$, $3 \times 5$, $5 \times 3$, $4 \times 2$, $2 \times 4$, $4 \times 1$, $1 \times 4$, $1 \times 2$, $2 \times 1$, $1 \times 3$.