Is it normal that scatterplot and residual (Anscombe) plot have the same shape but translated?

Question

I'm using R to produce a scatterplot and a residual (anscombe) plot. However I got something really weird. As you can see below.

Here is the scatter plot: And here is the residual:

As you can see they have the same exact shape, but they are just moved. Is this normal in a simple linear regression context?

Code

I am pretty confident of my code, but anyway, to take away any doubt this is a Minimal Example of my code:

# Plot the scatter plot, where x is the explanatory variable, y the response
plot(data$x, data$y)
# Fit linear model
data.lm <- lm( y ~x, data=data)
# Create standardized residuals
data.lm$sr <- rstandard(data.lm)
# Plot the Residual Plot
plot(data.lm$fitted.values, data.lm$sr)

Answer 1

They do not have the exact same shape. Three of the data points appear to be at about $(40,40),$ $(42,43),$ and $(44,42).$ Look at their images in the residual plot and you can see that the angles in the triangle have changed.

This is not a translation (which would preserve the shape) by a shear. When, as in this case, the correlation in the original scatterplot is close to $0$, then the change in shape will be small.

Now try this with these points: $(0,3) (0,6), (1,11), (1,12), (1,16).$ Look at the original plot and the residuals, and see what changes and what doesn't.

When you fit a straight line by least squares, the residual plot will always be a shear of the original scatterplot, for reasons that should become clear when you think about the relationship between the fitted line and the residuals.