This story covers how to install parquet tile on the diagonal.
Begin your diagonal layout in the same fashion as a perpendicular installation. Snap chalk lines at the midpoints of opposite walls and square the lines with a 3-4-5 triangle. From the center point, mark an equal distance on all four lines and connect the marks. Snap chalk lines from the corners through the midpoints on the connecting lines. Dry-lay the tiles along the diagonal so you'll achieve edge tiles of equal width.
Starting at the intersection of the layout lines, spread and comb mortar along one of the diagonal quadrants. Lay the first tile square against the intersection of the lines, using the edge of the tile (not the tongue or groove) as your reference plane. With the second tile slightly raised, engage the tongue in the groove of the first and press down and in. Continue setting tiles using this technique, one quadrant at a time, making diagonal cuts at the walls.