In an attempt to understand the quirks in our list of rotopes, I have created this tree:

I'm using | to mean a normal letter, and ' to mean a superscript letter.

Now, if you look at the first three dimensions, the number of rotopes is a power of 3: 1, 3, 9. However, in the 4th dimension, there are two new shapes: the duocylinder and the tiger, highlighted in red. The reason that analogies of these do not occur in lower dimensions is that they only have effect two dimensions later and if you add a (||) to a 1D |, you just get |(||) which is the same as the cylinder.

I'll look further into these, for higher dimensions.