In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L^p$ for $1 < p < \infty$, $p \neq 2$. We extend their construction to rearrangement-invariant function spaces and Hardy spaces. For any separable rearrangement-invariant space $X$ with nontrivial Boyd indices, we show that these subspaces are complemented (previously obtained by Ghawadrah via a different argument) and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to $X$. To this end, we introduce an ordinal index that detects whether $X$ embeds complementably into an arbitrary separable Banach space. Moreover, we give examples of spaces close to $L^1$ and $H^1$ where these subspaces fail to be complemented.
Based on joint work with Konstantinos Konstantos and Pavlos Motakis.