Primary sensory brain areas contain innate stereotyped topographies like retinotopy and somatotopy, but evidence of similar topographies underlying abstract cortical representations remains equivocal. Experience-driven self-organization provides a complementary mechanism for neural circuit development which instead yields conserved features like orientation tuning, but has also mainly been studied in sensory cortices or artificial neural networks (ANNs). Here we pursue a high level perspective on conservation of topographies and feature spaces throughout the cortex using blood oxygen level dependent fMRI across diverse cognitive, motor and sensory tasks from the Human Connectome Project (HCP). We quantify this by comparing regional spatial similarity of evoked responses to the similarity of representational geometry across participants. The latter measures response discriminability, and characterizes information a hypothetical downstream neuron can read out. Notably, conserved geometry appears through convergent learning in deep layers of otherwise diverse ANNs. We show interindividual topographic similarity rapidly decays along \textit{a priori} unimodal to transmodal gradients of cortical organization, a measure of polysynaptic "depth". Conversely, representational geometry does not show this relationship with cortical hierarchy. Only early sensorimotor regions show conserved topographies, but deep transmodal association areas show similar representations despite highly idiosyncratic organization. We further leverage family structure in the HCP to explore the effects of genes vs. self-organization and learning on cortical topographies and representations. We find evidence of heritable topographies, but not representational geometry, which is consistent with learned features, but surprisingly doesn't depend on common early life experiences and instead varies in an idiosyncratic manner. This pattern replicates findings in ANNs where input and output maps are hard coded while idiosyncratically tuned deep layers learn convergent feature spaces, but raises questions regarding the role of critical-period vs. continuous adult learning in the deepest biological circuits.