Also consider calling the rotated square a biased square or a square on the bias, where bias imparts the sense of diagonal, oblique, slanting.
For example, on the bias ordinarily refers to a 45° slope (or more precisely, the slope of a series of intersections of warp and weft) in sewing. From Merriam Webster:
a line diagonal to the grain of a fabric; especially: a line at a 45 degree angle to the selvage often utilized in the cutting of garments for smoother fit.
From OED1, sense A.1. of bias:
Slanting, oblique. Bias line : (in early geometry)
a diagonal or hypotenuse. [eg] 1551 RECORDE Pathw. Knowl. II. xxxii, By the Bias line, I meane that lyne, whiche in any square figure dooth runne from corner to corner.
You might also refer to an indexed square. Like rotated square, the phrase indexed square does not imply a certain angle, such as 45°; but on the other hand, indexing is not free rotation. It frequently is to cardinal angles or to multiples of 15, 30, or 45°. In short, indexing an item means rotating or moving it to a preset location:
Usually when the word indexing is used, it refers specifically to rotation. ... For example, Machinery’s Handbook, 25th edition, in its section on milling machine indexing, says, “Positioning a workpiece at a precise angle or interval of rotation for a machining operation is called indexing.”