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Case 1:   txy > 0 and sx > sy
The principal axes are counterclockwise to the current axes (because txy > 0) and no more than 45º away (because sx > sy).

Case 2:   txy < 0 and sx > sy
The principal axes are clockwise to the current axes (because txy < 0) and no more than 45º away (because sx > sy).

Case 3:   txy > 0 and sx < sy
The principal axes are counterclockwise to the current axes (because txy > 0) and between 45º and 90º away (because sx < sy).

Case 4:   txy < 0 and sx < sy
The principal axes are clockwise to the current axes (because txy < 0) and between 45º and 90º away (because sx < sy).

Case 5:   txy = 0 and sx > sy
The principal axes are aligned with the current axes (because sx > sy and txy = 0).

Case 6:   txy = 0 and sx < sy
The principal axes are exactly 90° from the current axes (because sx < sy and txy = 0).