The horizontal orientation angle and vertical inclination angle of an elongated subsurface object are key parameters for object identification and imaging in ground penetration radar (GPR) applications. Conventional methods can only extract the horizontal orientation angle or estimate both angles in narrow ranges due to limited polarimetric information and detection capability. To address these issues, this letter, for the first time, explores the possibility of leveraging neural networks with multi-polarimetric GPR data to estimate both angles of an elongated subsurface object in the entire spatial range. Based on the polarization-sensitive characteristic of an elongated object, we propose a multi-polarization aggregation and selection neural network (MASNet), which takes the multi-polarimetric radargrams as inputs, integrates their characteristics in the feature space, and selects discriminative features of reflected signal patterns for accurate orientation estimation. Numerical results show that our proposed MASNet achieves high estimation accuracy with an angle estimation error of less than 5{\deg}, which outperforms conventional methods by a large margin. The promising results obtained by the proposed method encourages one to think of new solutions for GPR related tasks by integrating multi-polarization information with deep learning techniques.