Palmprint biometrics garner heightened attention in palm-scanning payment and social security due to their distinctive attributes. However, prevailing methodologies singularly prioritize texture orientation, neglecting the significant texture scale dimension. We design an innovative network for concurrently extracting intra-scale and inter-scale features to redress this limitation. This paper proposes a scale-aware competitive network (SAC-Net), which includes the Inner-Scale Competition Module (ISCM) and the Across-Scale Competition Module (ASCM) to capture texture characteristics related to orientation and scale. ISCM efficiently integrates learnable Gabor filters and a self-attention mechanism to extract rich orientation data and discern textures with long-range discriminative properties. Subsequently, ASCM leverages a competitive strategy across various scales to effectively encapsulate the competitive texture scale elements. By synergizing ISCM and ASCM, our method adeptly characterizes palmprint features. Rigorous experimentation across three benchmark datasets unequivocally demonstrates our proposed approach's exceptional recognition performance and resilience relative to state-of-the-art alternatives.