Detecting a shallowly buried and elongated object and estimating its orientation using a commonly adopted co-polarized GPR system is challenging due to the presence of strong ground clutter that masks the target reflection. A cross-polarized configuration can be used to suppress ground clutter and reveal the object reflection, but it suffers from inconsistent detection capability which significantly varies with different object orientations. To address this issue, we propose a dual-cross-polarized detection (DCPD) method which utilizes two cross-polarized antennas with a special arrangement to detect the object. The signals reflected by the object and collected by the two antennas are combined in a rotationally invariant manner to ensure both effective ground clutter suppression and consistent detection irrespective of the object orientation. In addition, we present a dual-cross-polarized orientation estimation (DCPOE) algorithm to estimate the object orientation from the two cross-polarized data. The proposed DCPOE algorithm is less affected by environmental noise and performs robust and accurate azimuth angle estimation. The effectiveness of the proposed techniques in the detection and orientation estimation and their advantages over the existing method have been demonstrated using experimental data. Comparison results show that the maximum and average errors are 22.3{\deg} and 10.9{\deg} for the Alford rotation algorithm, while those are 4.9{\deg} and 1.8{\deg} for the proposed DCPOE algorithm in the demonstrated shallowly buried object cases. The proposed techniques can be unified in a framework to facilitate the investigation and mapping of shallowly buried and elongated targets.