Are there hyperbolas in the scatter plots of inversive congruential pseudorandom numbers?

Given a scatter plot of overlapping pairs (xn,xn+1) of pseudorandom numbers and the information that it stems from either a linear congruential or an inversive congruential generator with small modulus, it is usually no problem to guess the type of the generator: the points of a linear generator form a grid or lattice whereas inversive and explicit inversive congruential pseudorandom numbers exhibit hyperbola-like structures. We give an analytical description of this most eye-catching structural element of inversive congruential generators.