Despite the diversity of neural responses, the brain is highly structured both at anatomical and functional levels. Neurons in different brain areas exhibit different response profiles (large-scale anatomical organization), and within brain areas, the neurons can sometimes be grouped together into specialized subpopulations (categorical representations). Organization can also be found at the level of the representational geometry in the activity space, typically in the form of low-dimensional structure (e.g. disentangled abstract representations). However, it is unclear how the geometry in the activity space and the structure of the response profiles of individual neurons are related. Here, we systematically analyzed the geometric and selectivity structure of the neural population from 40+ cortical regions in mice performing a decision-making task (IBL Brainwide Map data set). We developed a reduced-rank regression model for studying the temporal and selectivity components of neuronal responses. We found that: 1) within each brain area, the neuronal response profiles are very diverse and weakly clustered (i.e., categorical) only in primary sensory areas; 2) clustering decreases along the cortical hierarchy; 3) when multiple brain areas are considered, we observe clustering that reflects the brain’s large scale organization. We then developed a mathematical theory relating clustering and maximal embedding dimensionality, predicting that they should be inversely correlated. This relationship was empirically verified as the embedding dimensionality of representations increased along the cortical hierarchy and was close to maximal. Consistently, the shattering dimensionality, which measures how many different classifications in the activity space can be solved by a linear readout, was close to the maximum (>90\%) across all areas, allowing maximum readout flexibility. These results provide a new mathematical and empirical perspective on selectivity and representation geometry in the neural code, suggesting that the diversity of neural responses allows for maximal dimensionality in all brain areas.