One useful fact about the residuals of the residuals of the y ~ x and the x ~ y regressions is that their correlation has opposite sign than the correlation between y and x. More precisely of r = corr(x, y) then the correlation between the two sets of residuals is r(1-r^2).

One useful fact about the residuals of the OLS Y ~ X and the X ~ Y regressions is that their correlation is then opposite of the correlation between Y and X.

Write Y = a1 + b1 X + e1 and X = a2 + b2 Y + e2.

If we denote r=corr(X,Y) the OLS estimates are

b1 = r * sY / sX

and

b2 = r * sX / sY

where sX is the standard deviation of X and sY the standard deviation of Y.

Then

Cov(e1,e2) = - r  (1-r^2) sX sY, Var(e1) = sY^2 (1-r^2), Var(e2) = sX^2 (1-r^2),

so Cor(e1,e2)= -r!