Causality Help!?!?

I know it is tempting to draw conclusions from the coefficients in regression models, caution is really needed. The coefficient values can be influenced by the scale of the variables. In your case, comparing coefficients between models with different dependent and independent variables isn't straightforward.

Additionally, the important disticnction you need to make is that the p-values indicate the statistical significance of each coefficient, not the strength of the effect. A low p-value suggests that you can reject the null hypothesis that the coefficient is zero, but it doesn't quantify the size of the effect.

Rather than directly comparing coefficients, consider standardizing your variables (subtracting the mean and dividing by the standard deviation) to make them comparable on a common scale. This allows for a more meaningful comparison of the magnitude of the effects.

From the t-stat and p-values in both models you can tell that:

First model:
First X-variable is not statistically significant! (95% confidence interval requires t-stat of >1.96)

Second model:
First X-variable (former Y) is not statistically significant! (95% confidence interval requires t-stat of >1.96)

The fact that the original X variable had a p-value of 0.10 suggests it might not be statistically significant at the conventional level. While the original X variable seems to have a larger effect (based on magnitude), its p-value raises questions about its statistical significance.

It appears that the intercept is significant in predicting the independent variable. You might also want to try re-estimating using one x-variable and the intercept and then assess the results.

Remember that the coefficients in a regression model indicate the change in the dependent variable for a one-unit change in the independent variable, holding other variables constant. The sign of the coefficient tells you the direction of the relationship, and the magnitude tells you the size of the effect.

You cannot directly conclude that one variable has "twice as much effect" based solely on the coefficient values. The scale of the variables matters. If you've standardized your variables, you can directly compare the magnitudes of the coefficients. A coefficient of 0.90 in one model and 0.49 in another would suggest a larger effect in the former.

Given the standard scale, a coefficient of 0.90 in the original model implies that, on average, a one-unit increase in X1​ is associated with a 0.90-unit increase in Y. Similarly, a coefficient of 0.49 in the swapped model implies that, on average, a one-unit increase in the new X1​ (formerly Y) is associated with a 0.49-unit increase in X1​.

Always, consider the context of your study, the nature of the variables, and the goals of your analysis when interpreting these kinds of results. It might be beneficial to possibly conduct additional analyses to validate your findings, as I mentioned above.

Keep in mind that "domain knowledge" is crucial, and statistical results should be interpreted in conjunction with your expertise. The expertise is what comes first, in my opinion.