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With increasing promise of radiomics and deep learning approaches in capturing subtle patterns associated with disease response on routine MRI, there is an opportunity to more closely combine components from both approaches within a single architecture. We present a novel approach to integrating multi-scale, multi-oriented wavelet networks (WN) into a convolutional neural network (CNN) architecture, termed a deep hybrid convolutional wavelet network (DHCWN). The proposed model comprises the wavelet neurons (wavelons) that use the shift and scale parameters of a mother wavelet function as its building units. Whereas the activation functions in a typical CNN are fixed and monotonic (e.g. ReLU), the activation functions of the proposed DHCWN are wavelet functions that are flexible and significantly more stable during optimization. The proposed DHCWN was evaluated using a multi-institutional cohort of 153 pre-treatment rectal cancer MRI scans to predict pathologic response to neoadjuvant chemoradiation. When compared to typical CNN and a multilayer wavelet perceptron (DWN-MLP) 2D and 3D architectures, our novel DHCWN yielded significantly better performance in predicting pathologic complete response (achieving a maximum accuracy of 91.23% and a maximum AUC of 0.79), across multi-institutional discovery and hold-out validation cohorts. Interpretability evaluation of all three architectures via Grad-CAM and Shapley visualizations revealed DHCWNs best captured complex texture patterns within tumor regions on MRI as associated with pathologic complete response classification. The proposed DHCWN thus offers a significantly more extensible, interpretable, and integrated solution for characterizing predictive signatures via routine imaging data.