This paper defines an alternative notion, described as data-based, of geometric quantiles on Hadamard spaces, in contrast to the existing methodology, described as parameter-based. In addition to having the same desirable properties as parameter-based quantiles, these data-based quantiles are shown to have several theoretical advantages related to large-sample properties like strong consistency and asymptotic normality, breakdown points, extreme quantiles and the gradient of the loss function. Using simulations, we explore some other advantages of the data-based framework, including simpler computation and better adherence to the shape of the distribution, before performing experiments with real diffusion tensor imaging data lying on a manifold of symmetric positive definite matrices. These experiments illustrate some of the uses of these quantiles by testing the equivalence of the generating distributions of different data sets and measuring distributional characteristics.