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In class-incremental learning, the model is expected to learn new classes continually while maintaining knowledge on previous classes. The challenge here lies in preserving the model's ability to effectively represent prior classes in the feature space, while adapting it to represent incoming new classes. We propose two distillation-based objectives for class incremental learning that leverage the structure of the feature space to maintain accuracy on previous classes, as well as enable learning the new classes. In our first objective, termed cross-space clustering (CSC), we propose to use the feature space structure of the previous model to characterize directions of optimization that maximally preserve the class: directions that all instances of a specific class should collectively optimize towards, and those that they should collectively optimize away from. Apart from minimizing forgetting, this indirectly encourages the model to cluster all instances of a class in the current feature space, and gives rise to a sense of herd-immunity, allowing all samples of a class to jointly combat the model from forgetting the class. Our second objective termed controlled transfer (CT) tackles incremental learning from an understudied perspective of inter-class transfer. CT explicitly approximates and conditions the current model on the semantic similarities between incrementally arriving classes and prior classes. This allows the model to learn classes in such a way that it maximizes positive forward transfer from similar prior classes, thus increasing plasticity, and minimizes negative backward transfer on dissimilar prior classes, whereby strengthening stability. We perform extensive experiments on two benchmark datasets, adding our method (CSCCT) on top of three prominent class-incremental learning methods. We observe consistent performance improvement on a variety of experimental settings.