The presence of spherical distortion on the Equirectangular image is an acknowledged challenge in dense regression computer vision tasks, such as surface normal estimation. Recent advances in convolutional neural networks (CNNs) strive to mitigate spherical distortion but often fall short in capturing holistic structures effectively, primarily due to their fixed receptive field. On the other hand, vision transformers (ViTs) excel in establishing long-range dependencies through a global self-attention mechanism, yet they encounter limitations in preserving local details. We introduce \textit{PanoNormal}, a monocular surface normal estimation architecture designed for 360{\deg} images, which combines the strengths of CNNs and ViTs. Specifically, we employ a multi-level global self-attention scheme with the consideration of the spherical feature distribution, enhancing the comprehensive understanding of the scene. Our experimental results demonstrate that our approach achieves state-of-the-art performance across multiple popular 360{\deg} monocular datasets. The code and models will be released.