We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on $Rp$ and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are $K$-homothetic (i.e.~scalar multiples of a convex body $K\subseteq Rp$). When $K$ is known, we prove that the $K$-homothetic log-concave maximum likelihood estimator based on $n$ independent observations from such a density achieves the minimax optimal rate of convergence with respect to, e.g., squared Hellinger loss, of order $n-4/5$, independent of~$p$. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst-case and adaptive risk bounds in cases where $K$ is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when $n$ and $p$ are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data.