As Xi'an noted in his answer the $\beta$ and $\gamma$ are related to each other by relating to the conditional means $X|Y$ and $Y|X$ (which in their turn relate to a single joint distribution) these are not symmetric in the sense that $\beta = 1/\gamma$. This is neither the case if you would have the true $\sigma$ and $\rho$ instead of estimates. You have $$\beta = \rho_{XY} \frac{\sigma_Y}{\sigma_X}$$ and $$\gamma = \rho_{XY} \frac{\sigma_X}{\sigma_Y}$$
(see also simple linear regression on wikipedia)
But, you may wonder whether these conditional probabilities and regression lines is what you need to determine your ratios of $X$ and $Y$. It is unclear to me how you would wish to use a regression line in the computation of an optimal ratio.
Below is an alternative way to compute the ratio. This method does have symmetry (ie if you switch X and Y then you will get the same ratio).
Say, the yields of bonds $X$ and $Y$ are distributed according to a multivariate normal distribution$^\dagger$ with correlation $\rho_{XY}$ and standard deviations $\sigma_X$ and $\sigma_Y$ then the yield of a hedge that is sum of $X$ and $Y$ will be normal distributed:
$$H = \alpha X + (1-\alpha) Y \sim N(\mu_H,\sigma_H^2)$$
were $0 \leq \alpha \leq 1$ and with
$$\begin{array}{rcl} \mu_H &=& \alpha \mu_X+(1-\alpha) \mu_Y \\ \sigma_H^2 &=& \alpha^2 \sigma_X^2 + (1-\alpha)^2 \sigma_Y^2 + 2 \alpha (1-\alpha) \rho_{XY} \sigma_X \sigma_Y \\ & =& \alpha^2(\sigma_X^2+\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y) + \alpha (-2 \sigma_Y^2+2\rho_{XY}\sigma_X\sigma_Y) +\sigma_Y^2 \end{array} $$
The mean $\mu_H$ will be maximal for $$\alpha = 0 \text{ or } \alpha=1$$ or not existing when $\mu_X=\mu_Y$.
The variance $\sigma_H^2$ will be minimal for $$\alpha = 1 - \frac{\sigma_X^2 -\rho_{XY}\sigma_X\sigma_Y}{\sigma_X^2 +\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y} = \frac{\sigma_Y^2-\rho_{XY}\sigma_X\sigma_Y}{\sigma_X^2+\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y} $$
The optimum will be somewhere in between those two extremes and depends on how you wish to compare losses and gains, but note that now there is a symmetry between $\alpha$ and $1-\alpha$. It does not matter whether you use the hedge $H=\alpha X+(1-\alpha)Y$ or the hedge $H=\beta Y + (1-\beta) X$. You will get the same ratios in terms of $\alpha = 1-\beta$.
Improvements of the model can be made by using different distributions than multivariate normal. Also you could incorporate the time in a more sophisticated model to make better predictions of future values for the pair $X,Y$.
$\dagger$ This is a simplification but it suits the purpose of explaining how one can, and should, perform the analysis to find an optimal ratio without a regression line.