  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 \'etale 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 $\mathcal{O}_2^p\cong \mathcal{O}_2^p\otimes\mathcal{O}_2^p$
preserves the canonical Cartan subalgebras.
