For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\mathrm{Iso}_0(X)\curvearrowright S_X$.

When $X=L^p([0,1])$ the group $\mathrm{Iso}_0(X)$ is Polish, so we can ask how complex the orbit equivalence relation induced on $S_X$ is in the descriptive set theoretic sense. By a result of Pełczyński and Rolewicz every orbit is dense, but I don´t know much apart from this.

Are orbits meagre? Does this equivalence relation admit classification by countable structures? What are good references for this kind of questions and for more general questions along those lines (for example when $X$ is the Gurarij space)?