I believe the problem is that (unless R^2 = 1) the lines are not y = mx + c and x = ny +d but y = mx + c + e1 and x = ny + d + e2, where the e are errors.

If you have R you can see this this way

x <- c(1,3,6,7,9)

y <- 3*x + 2

m1 <- lm(y~x)

m2 <- lm(x~y)

coef(m1)

coef(m2)

where a strict relationship holds. But add some error

y2 <- 3*x + 2 + rnorm(5)

and re-run

m3 <- lm(y2~x)

m4 <- lm(x~y2)

and the relationship fails