Recognizing interval graphs--"equivalent intervals"
Imagine scanning the real line from left to right. Whenever an interval starts or ends, you make notice of it. For example, perhaps at some representation, at the point $3$ the intervals for $x_2,x_7$ start and the interval for $x_{15}$ ends, and at the point $4$ the interval for $x_2$ ends, and no points occur in between.

Suppose we "remove" the portion $(3.25,3.75)$ of the real line, thus pushing the point $4$ back to $3.5$. This only affects the value of some ordinates, so we would like to say that these two interval representations are "the same". Indeed, they are equivalent under the definition you give, since if we make a list of which intervals start and end at what point, then the lists would be identical, the only difference being the value of the ordinates.

Another operation we can do is reflecting the real line, say around zero. Now the interval for $x_2$ is $(-4,-3)$ instead of $(3,4)$, so that at the point $-4$ the interval $x_2$ starts, and at the point $-3$ the intervals for $x_2,x_7$ end and the interval for $x_{15}$ starts. Again, we would like both of these representation to count as "the same". This transformation is captured by the reversal clause in the definition you quote.