Do bitwise rotations and increments modulo $2^n$ generate the symmetric group?

Question

Let's say we have two operations we can perform on a binary number of length $n$:

1. Right-rotation, where the most significant bit is taken off and inserted in the one's place, pushing all other digits up.
2. Adding one to the number modulo $2^n$.

Do these two operations together generate the symmetric group? In other words, can every bijective function $\mathbb{Z}/2^n\mathbb{Z} \rightarrow \mathbb{Z}/2^n\mathbb{Z}$ be made through some composition of some number of those two operations? I don't see an easy way to prove it can't be done, because they don't distribute over each other, commute, or have an obvious rule about which bits can or can't impact other bits.