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Controlled table-to-text generation seeks to generate natural language descriptions for highlighted subparts of a table. Previous SOTA systems still employ a sequence-to-sequence generation method, which merely captures the table as a linear structure and is brittle when table layouts change. We seek to go beyond this paradigm by (1) effectively expressing the relations of content pieces in the table, and (2) making our model robust to content-invariant structural transformations. Accordingly, we propose an equivariance learning framework, which encodes tables with a structure-aware self-attention mechanism. This prunes the full self-attention structure into an order-invariant graph attention that captures the connected graph structure of cells belonging to the same row or column, and it differentiates between relevant cells and irrelevant cells from the structural perspective. Our framework also modifies the positional encoding mechanism to preserve the relative position of tokens in the same cell but enforce position invariance among different cells. Our technology is free to be plugged into existing table-to-text generation models, and has improved T5-based models to offer better performance on ToTTo and HiTab. Moreover, on a harder version of ToTTo, we preserve promising performance, while previous SOTA systems, even with transformation-based data augmentation, have seen significant performance drops. Our code is available at https://github.com/luka-group/Lattice.