Divide a square into rectangles where each occurs once in each orientation
Solution 1:
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: $3 \times 1$, $3 \times 5$, $5 \times 3$, $4 \times 2$, $2 \times 4$, $4 \times 1$, $1 \times 4$, $1 \times 2$, $2 \times 1$, $1 \times 3$.