Let $X$ and $Y$ be mean 0 and variance 1 random variables such that we choose $\alpha$ and $\beta$ to minimise
$$\mathbb{E}(X-\beta Y)^2$$
and
$$\mathbb{E}(Y-\alpha X)^2$$
after not so difficult derivation, I arrive at $\alpha = \mathbb{E}(XY)/\mathbb{E}X^2$ and $\beta = \mathbb{E}(XY)/\mathbb{E}Y^2$, so $\alpha = \beta$.
This seems very strange, because if $y=mx$ is regression line, then surely $x = \frac{1}{m }y$.