Let $R\subset\mathbb{R}^2$ be a normal domain, normal with respect to the x-axis (the definition I'm using)
Let $\gamma:\mathbb{R}^2\to\mathbb{R}^2$ be a rotation.
Then:
A. $\gamma(R)$ is normal with respect to the x-axis
B. $\gamma(R)$ can be normal with respect to the y-axis
C. $\gamma(R)$ is normal with respect to the x-axis or to the y-axis
D. $\gamma(R)$ can be normal with respect to the x-axis and to the y-axis even if $R$ was normal with respect to the x-axis only
The half-annulus is a simple example showing that A is false and B is true.
I would guess C-true and D-false, but I have no proof for my suspicions.
C. The same $U$-shaped half-annulus rotated by $\pi/4$ with respect to its center is not normal with respect to the $x$-axis and to the $y$-axis.
D. A $V$-shape is normal with respect to the $x$-axis only, but rotated to $L$-shape it becomes normal with respect to the $x$-axis and to the $y$-axis.