For $p\in [1,\infty)$, we show that every unital $L^p$-operator algebra contains a unique maximal C*-subalgebra, which is always abelian if $p\neq 2$. Using this, we canonically associate to every unital $L^p$-operator algebra $A$ an étale groupoid $\mathcal{G}_A$, which in many cases of interest is a complete invariant for $A$. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case $p=2$; the most striking one being that of crossed products by topologically free actions.

Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the $L^p$-analog $\mathcal{O}_2^p$ of the Cuntz algebra, there is no isometric isomorphism between $\mathcal{O}_2^p$ and $\mathcal{O}_2^p\otimes^p\mathcal{O}_2^p$, when $p\neq 2$. In particular, we deduce that there is no $L^p$-version of Kirchberg's absorption theorem, and that there is no $K$-theoretic classification of purely infinite simple amenable $L^p$-operator algebras for $p\neq 2$. Our methods also allow us to recover a folklore fact in the case of $C^*$-algebras ($p=2$), namely that no isomorphism $\mathcal{O}_2\cong \mathcal{O}_2\otimes\mathcal{O}_2$ preserves the canonical Cartan subalgebras.