Neural reconstructions often trade structure for fidelity, yielding dense and unstructured meshes with irregular topology and weak part boundaries that hinder editing, animation, and downstream asset reuse. We present DualPrim, a compact and structured 3D reconstruction framework. Unlike additive-only implicit or primitive methods, DualPrim represents shapes with positive and negative superquadrics: the former builds the bases while the latter carves local volumes through a differentiable operator, enabling topology-aware modeling of holes and concavities. This additive-subtractive design increases the representational power without sacrificing compactness or differentiability. We embed DualPrim in a volumetric differentiable renderer, enabling end-to-end learning from multi-view images and seamless mesh export via closed-form boolean difference. Empirically, DualPrim delivers state-of-the-art accuracy and produces compact, structured, and interpretable outputs that better satisfy downstream needs than additive-only alternatives.