The generalization of deep learning-based low-dose computed tomography (CT) reconstruction models to doses unseen in the training data is important and remains challenging. Previous efforts heavily rely on paired data to improve the generalization performance and robustness through collecting either diverse CT data for re-training or a few test data for fine-tuning. Recently, diffusion models have shown promising and generalizable performance in low-dose CT (LDCT) reconstruction, however, they may produce unrealistic structures due to the CT image noise deviating from Gaussian distribution and imprecise prior information from the guidance of noisy LDCT images. In this paper, we propose a noise-inspired diffusion model for generalizable LDCT reconstruction, termed NEED, which tailors diffusion models for noise characteristics of each domain. First, we propose a novel shifted Poisson diffusion model to denoise projection data, which aligns the diffusion process with the noise model in pre-log LDCT projections. Second, we devise a doubly guided diffusion model to refine reconstructed images, which leverages LDCT images and initial reconstructions to more accurately locate prior information and enhance reconstruction fidelity. By cascading these two diffusion models for dual-domain reconstruction, our NEED requires only normal-dose data for training and can be effectively extended to various unseen dose levels during testing via a time step matching strategy. Extensive qualitative, quantitative, and segmentation-based evaluations on two datasets demonstrate that our NEED consistently outperforms state-of-the-art methods in reconstruction and generalization performance. Source code is made available at https://github.com/qgao21/NEED.