This paper revisits the performance of Rademacher random projections, establishing novel statistical guarantees that are numerically sharp and non-oblivious with respect to the input data. More specifically, the central result is the Schur-concavity property of Rademacher random projections with respect to the inputs. This offers a novel geometric perspective on the performance of random projections, while improving quantitatively on bounds from previous works. As a corollary of this broader result, we obtained the improved performance on data which is sparse or is distributed with small spread. This non-oblivious analysis is a novelty compared to techniques from previous work, and bridges the frequently observed gap between theory and practise. The main result uses an algebraic framework for proving Schur-concavity properties, which is a contribution of independent interest and an elegant alternative to derivative-based criteria.