Self-supervised learning methods based on image patch reconstruction have witnessed great success in training auto-encoders, whose pre-trained weights can be transferred to fine-tune other downstream tasks of image understanding. However, existing methods seldom study the various importance of reconstructed patches and the symmetry of anatomical structures, when they are applied to 3D medical images. In this paper we propose a novel Attentive Symmetric Auto-encoder (ASA) based on Vision Transformer (ViT) for 3D brain MRI segmentation tasks. We conjecture that forcing the auto-encoder to recover informative image regions can harvest more discriminative representations, than to recover smooth image patches. Then we adopt a gradient based metric to estimate the importance of each image patch. In the pre-training stage, the proposed auto-encoder pays more attention to reconstruct the informative patches according to the gradient metrics. Moreover, we resort to the prior of brain structures and develop a Symmetric Position Encoding (SPE) method to better exploit the correlations between long-range but spatially symmetric regions to obtain effective features. Experimental results show that our proposed attentive symmetric auto-encoder outperforms the state-of-the-art self-supervised learning methods and medical image segmentation models on three brain MRI segmentation benchmarks.