Regression to the mean I recently asked this question 

In regression model with random regressors
$$(1) \ \ y = a + bx + e$$
can I change the equation to
$$(2) \ \  x = (-a/b) + (1/b)y + (-1/b)e$$
and consistently estimate $(1/b)$ with OLS?

which is already answered here whats the difference between regression of x on y versus y on x and related here. Very interesting post but I would like to ask the following follow up question:
I guess we can all agree that (1) and (2) are mathematically equivalent. I then noticed that if I simulate equation (1) by 


*

*Drawing x and drawing e with x independently of x and then calulate y using formula (1) the OLS estimator regressing y on x consistently estimates a and b.


However if simulate by


*Drawing y and drawing e indenpently and the calculate x according to (2) then the OLS estimator regressing x on y consistently estimates $\lambda_0=(-a/b)$ and $\lambda_1 := 1/b$. 


The question then is does this not mean that the model statement (1) is somehow incomplete in the sense that a more complete model statement would be 
$$y = a + b x + e \wedge x \perp e$$ versus
$$y = a + b x + e \wedge y \perp e$$
because for the latter model it would exactly be necessary to use OLS as associated with the equation 
$$(2) \ \  x = (-a/b) + (1/b)y + (-1/b)e$$
to get consistent estimates. So my question boils down to when we simply write $$y = a + bx + e$$ the statement is somewhat incomplete missing the variable dependencies?
 A: The issue here is that equation $(1)$ does not fully specify the regression model, so yes, that equation is an incomplete statement of the model.  In order to fully specify the regression model, you need to specify the distribution of $e$ conditional on $x$.  For a homoscedastic Gaussian linear regression model, the defining equations are:
$$y = a + b x + e
\quad \quad \quad \quad \quad
e | x \sim \text{N}(0, \sigma^2).$$
The second equation does indeed imply that $e \perp x$, but it is not generally true that $e \perp y$.  Thus, while you can certainly manipulate equation $(1)$ into $(2)$ (so long as $b \neq 0$), this is not sufficient to give you a linear regression model for $x$ in terms of $y$.  This is what whuber is referring to in the comments when he says that the regression model is inherently asymmetric.
It is possible to obtain a symmetric case by making the much stronger assumption that $(x,y)$ are jointly normally distributed.  In this case it is possible to write either variable in terms of the other using the form of a linear regression model.  This assumption is stronger than the assumptions of a linear regression model on one variable, which only specifies the conditional distribution of that variable given the other variable.
