The 2D Gaussian distribution is known for its rotational symmetry (as shown in this 3blue1brown video), and it's also known that the radius $R = \sqrt{X^2 + Y^2}$ in polar coordinates follows a Rayleigh distribution. Let me try to put these valid facts against each other towards a contradiction:
Consider $p(R\mid\theta)$ for a given $\theta\in(-\pi,\pi)$. From the rotational symmetry, we know that if we rotate the bell, we still get the normal distribution, so the radius, which is constrained to be positive, must follow a half-normal distribution. For instance, if $\theta=0$, we would be considering the distribution of the radius $R$ constrained to the positive x axis, which is basically $X|Y=0,X>0$. However, we know that the radius $R$ follows a Rayleigh distribution regardless of the angle, not a half-normal. Absurd.
Where is the flaw in this argument and why isn't it so obvious?
I had a similar question here, and the solution is yet another example of Borel's paradox. The conditional probability $P_{XY|\Theta}(x, y | \theta)$ is a half-normal distribution within some Cartesian strip, but the conditional probability $P_{R|\Theta}(r, \theta)$ is a Rayleigh distribution within some wedge of $\theta$.