Diffusion magnetic resonance imaging (dMRI) is pivotal for probing the microstructure of the rapidly-developing fetal brain. However, fetal motion during scans and its interaction with magnetic field inhomogeneities result in artifacts and data scattering across spatial and angular domains. The effects of those artifacts are more pronounced in high-angular resolution fetal dMRI, where signal-to-noise ratio is very low. Those effects lead to biased estimates and compromise the consistency and reliability of dMRI analysis. This work presents HAITCH, the first and the only publicly available tool to correct and reconstruct multi-shell high-angular resolution fetal dMRI data. HAITCH offers several technical advances that include a blip-reversed dual-echo acquisition for dynamic distortion correction, advanced motion correction for model-free and robust reconstruction, optimized multi-shell design for enhanced information capture and increased tolerance to motion, and outlier detection for improved reconstruction fidelity. The framework is open-source, flexible, and can be used to process any type of fetal dMRI data including single-echo or single-shell acquisitions, but is most effective when used with multi-shell multi-echo fetal dMRI data that cannot be processed with any of the existing tools. Validation experiments on real fetal dMRI scans demonstrate significant improvements and accurate correction across diverse fetal ages and motion levels. HAITCH successfully removes artifacts and reconstructs high-fidelity fetal dMRI data suitable for advanced diffusion modeling, including fiber orientation distribution function estimation. These advancements pave the way for more reliable analysis of the fetal brain microstructure and tractography under challenging imaging conditions.