Millimeter-wave (mmWave) radar has emerged as a compact and powerful sensing modality for advanced perception tasks that leverage machine learning techniques. It is particularly effective in scenarios where vision-based sensors fail to capture reliable information, such as detecting occluded objects or distinguishing between different surface materials in indoor environments. Due to the non-linear characteristics of mmWave radar signals, deep learning-based methods are well suited for extracting relevant information from in-phase and quadrature (IQ) data. However, the current state of the art in IQ signal-based occluded-object and material classification still offers substantial potential for further improvement. In this paper, we propose a bidirectional cross-attention fusion network that combines IQ-signal and FFT-transformed radar features obtained by distinct complex-valued convolutional neural networks (CNNs). The proposed method achieves improved performance and robustness compared to standalone complex-valued CNNs. We achieve a near-perfect material classification accuracy of 99.92% on samples collected at same sensor-to-surface distances used during training, and an improved accuracy of 67.38% on samples measured at previously unseen distances, demonstrating improved generalization ability across varying measurement conditions. Furthermore, the accuracy for occluded object classification improves from 91.99% using standalone complex-valued CNNs to 94.20% using our proposed approach.