Why use independence test and not normality test for structure identification?

Question

In the book 'Elements of causal inference', the code below is used an example for structure identification.

library(dHSIC)
library(mgcv)
X <- rnorm(200)
Y <- X^3 + rnorm(200)
modelforw <- gam(Y ~ s(X))
modelbackw <- gam(X ~ s(Y))
dhsic.test(modelforw$residuals, X)$p.value
dhsic.test(modelbackw$residuals, Y)$p.value

The example is clear, but my doubt is, why is he using an independence test and not a normality test? Given that the data is normal, wouldn't it be easier to test for that?

Answer 1

The answer is in section 4.2.1 Additive Noise Models, here's an excerpt:

In this method they only need to check the independence of residuals from $x\sim y$ and $y\sim x$ models from $y$ and $x$ to see whether there is an asymmetry, which tells them whether there is causality