Say I have a unitary $U = e^{-iHt}$ where $H = \alpha X + Z$.
First, suppose $U = I$. Then it rotates a set of initial states to themselves. Say I'm working on a computational basis, then on the Bloch sphere $U$ will rotate $\lvert 0 \rangle$ and $\lvert 1 \rangle$ to $\lvert 0 \rangle$ and $\lvert 1 \rangle$, respectively.
Now, consider $\alpha = 0$ so that $U = e^{-iZt}$. For any $t$ such that $U \neq I$, it physically does the same thing on the Bloch sphere: $U$ will rotate $\lvert 0 \rangle$ and $\lvert 1 \rangle$ to $\lvert 0 \rangle$ and $\lvert 1 \rangle$, respectively. But definitely, $U$ isn't the identity gate.
If someone gives me a description of $U$ such that "$U$ rotates each basis to itself", then as mentioned above $U$ is not necessarily $I$. So there should be something more than this description to characterize that $U = I$. What are these additional descriptions that I'm missing in terms of the Bloch sphere? It seems like the answer should be related to the relative phase between $U\lvert 0 \rangle$ and $U\lvert 1 \rangle$, but can we describe this phase on the Bloch sphere? Any insightful answer would be much appreciated.