A commonly studied type of linear function in geometric algebra (and more generally, exterior algebra) is the outermorphism. For reference, here's Wikipedia's definition:
Let $f$ be an $\Bbb R$-linear map from $V$ to $W$. The outermorphism of $f$ is the unique map $\underline{\mathsf{f}} : \Lambda(V) \to \Lambda(W)$ satisfying $$ \underline{\mathsf{f}}(x) = f(x)\\ \underline{\mathsf{f}}(A \wedge B) = \underline{\mathsf{f}}(A) \wedge \underline{\mathsf{f}}(B)\\ \underline{\mathsf{f}}(A + B) = \underline{\mathsf{f}}(A) + \underline{\mathsf{f}}(B)\\ \underline{\mathsf{f}}(1) = 1$$ for all vectors $x$ and all multivectors $A$ and $B$, where $\Lambda(V)$ denotes the exterior algebra over $V$.
Why do we not also study linear mappings which preserve the inner product: "innermorphisms"? We could define it analogously. The only change in the definition above would be the second equation would read $$\underline{\mathsf{f}}(A \cdot B) = \underline{\mathsf{f}}(A) \cdot \underline{\mathsf{f}}(B)$$
There are at least $2$ examples of these functions: both reflections and rotations preserve the inner product (and have all of the other properties listed above).
Is it that these are the only ones? Or that the only functions which are innermorphisms are also the orthogonal functions and thus we are already studying them?
I'm just not sure why outermorphisms are so useful (and they are), but that analogous functions which preserve the inner product are apparently not.
We do, but for historical reasons they are called (linear) isometries.