Tactile recognition of 3D objects remains a challenging task. Compared to 2D shapes, the complex geometry of 3D surfaces requires richer tactile signals, more dexterous actions, and more advanced encoding techniques. In this work, we propose TANDEM3D, a method that applies a co-training framework for exploration and decision making to 3D object recognition with tactile signals. Starting with our previous work, which introduced a co-training paradigm for 2D recognition problems, we introduce a number of advances that enable us to scale up to 3D. TANDEM3D is based on a novel encoder that builds 3D object representation from contact positions and normals using PointNet++. Furthermore, by enabling 6DOF movement, TANDEM3D explores and collects discriminative touch information with high efficiency. Our method is trained entirely in simulation and validated with real-world experiments. Compared to state-of-the-art baselines, TANDEM3D achieves higher accuracy and a lower number of actions in recognizing 3D objects and is also shown to be more robust to different types and amounts of sensor noise. Video is available at https://jxu.ai/tandem3d.