With the rapid growth of hyperspectral data archives in remote sensing (RS), the need for efficient storage has become essential, driving significant attention toward learning-based hyperspectral image (HSI) compression. However, a comprehensive investigation of the individual and joint effects of spectral and spatial compression on learning-based HSI compression has not been thoroughly examined yet. Conducting such an analysis is crucial for understanding how the exploitation of spectral, spatial, and joint spatio-spectral redundancies affects HSI compression. To address this issue, we propose Adjustable Spatio-Spectral Hyperspectral Image Compression Network (HyCASS), a learning-based model designed for adjustable HSI compression in both spectral and spatial dimensions. HyCASS consists of six main modules: 1) spectral encoder module; 2) spatial encoder module; 3) compression ratio (CR) adapter encoder module; 4) CR adapter decoder module; 5) spatial decoder module; and 6) spectral decoder module. The modules employ convolutional layers and transformer blocks to capture both short-range and long-range redundancies. Experimental results on three HSI benchmark datasets demonstrate the effectiveness of our proposed adjustable model compared to existing learning-based compression models, surpassing the state of the art by up to 2.36 dB in terms of PSNR. Based on our results, we establish a guideline for effectively balancing spectral and spatial compression across different CRs, taking into account the spatial resolution of the HSIs. Our code and pre-trained model weights are publicly available at https://git.tu-berlin.de/rsim/hycass .