Why is it valid to say that both a function and its inverse describe the same relationship

A comparison between a function and its inverse would show that the domain and range of the original function swap. The domain of the function becomes the range of the inverse, the range of the function becomes the domain of its inverse. 

Looking at ordered pairs of the function and its inverse would look like this:

(2,4) on the original function becomes (4,2) on the inverse. 

While the graph of a function and its inverse are noticeably different an important thing to note is that it is merely a reflection across the line y=x. 

So even though they appear different you are looking at the same relationship just as y vs. x instead of x vs. y