What happens to correlation coefficient if you swap the independent
and dependent variables?

Howdy;

So, if you plot data in a scatter plot, then run your best-fit
correlation analysis, you will get the same answer regardless of which
is plotted as your X and Y.

Correlation coefficient remains the same.

The correlation coefficient describes the closeness of the points on a
scatter diagram to the least squares regression line.  If the points
all lie on the least square regresion line and the line has a positive
gradient the correlation coefficient is 1.  Similarly if the points
all lie on the line and the line has a negative gradient the
coefficient is -1.  If the dependent and independent variables are
swopped, the points on the scatter diagram are reflected in the line
y=x.  This changes the position of the points and the line ( because
the lines position is dependent upon the points).  However the
relative positions between the points on the scatter diagram and the
least squares regression line is unchanged.  Therefore the correlation
coefficient is unchanged.

The coefficient will remain the same.
http://mathworld.wolfram.com/CorrelationCoefficient.html
Look at equation 17.  Notice how you can swap the x and y without
altering what it means.

I can see how one would think it wouldn't.  A least squares regression
calculates the residuals vertically (assuming error in the dependant
variable).  Swapping the variables would be like calculating them
horizontally (assuming error in the independant variable).  However,
correlation is based on covariance, which is symmetric.