Transformer emerges as an active research topic in medical image analysis. Yet, three substantial challenges limit the effectiveness of both 2D and 3D Transformers in 3D medical image classification: 1) Challenge in capturing spatial structure correlation due to the unreasonable flattening operation; 2) Challenge in burdening the high computational complexity and memory consumption due to the quadratic growth of computational complexity and memory consumption for 3D medical data; 3) Challenge in discriminative representation learning, due to data-sensitivity. To address the above challenges, a novel Cross-dimensional Transformer (CdTransformer) and a creative Class-consistent Contrastive Learning (CcCL) are proposed. Specifically, CdTransformer consists of two novel modules: 1) Cross-dimensional Attention Module (CAM), which breaks the limitation that Transformer cannot reasonably establish spatial structure correlation when meeting 3D medical data, meanwhile, reduces the computational complexity and memory consumption. 2) Inter-dimensional Feed-forward Network (IdFN), which addresses the challenge of traditional feed-forward networks not being able to learn depth dimension information that is unique to 3D medical data. CcCL innovatively takes full advantage of the inter-class and intra-class features from the slice-distorted samples to boost Transformer in learning feature representation. CdTransformer and CcCL are validated on six 3D medical image classification tasks. Extensive experimental results demonstrate that CdTransformer outperforms state-of-the-art CNNs and Transformers on 3D medical image classification, and CcCL enables significantly improving Transformer in discriminative representation learning.