  Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by
multiplicatively independent exponents, and suppose that either $F$ is not a
product set, or it is a product set with marginals of dimension strictly
between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq
F$ is an isometry composed of reflections about lines parallel to the axes. Our
approach utilizes the structure of tangent sets of $F$, obtained by "zooming
in" on points of $F$, projection theorems for products of self-similar sets,
and logarithmic commensurability type results for self similar sets in the
line.
