We study the task of Composed Image Retrieval (CoIR), where a query is composed of two modalities, image and text, extending the user's expression ability. Previous methods typically address this task by a separate encoding of each query modality, followed by late fusion of the extracted features. In this paper, we propose a new approach, Cross-Attention driven Shift Encoder (CASE), employing early fusion between modalities through a cross-attention module with an additional auxiliary task. We show that our method outperforms the existing state-of-the-art, on established benchmarks (FashionIQ and CIRR) by a large margin. However, CoIR datasets are a few orders of magnitude smaller compared to other vision and language (V&L) datasets, and some suffer from serious flaws (e.g., queries with a redundant modality). We address these shortcomings by introducing Large Scale Composed Image Retrieval (LaSCo), a new CoIR dataset x10 times larger than current ones. Pre-training on LaSCo yields a further performance boost. We further suggest a new analysis of CoIR datasets and methods, for detecting modality redundancy or necessity, in queries.