Dealing with tabular data is challenging due to partial information, noise, and heterogeneous structure. Existing techniques often struggle to simultaneously address key aspects of tabular data such as textual information, a variable number of columns, and unseen data without metadata besides column names. We propose a novel architecture, \textit{basis transformers}, specifically designed to tackle these challenges while respecting inherent invariances in tabular data, including hierarchical structure and the representation of numeric values. We evaluate our design on a multi-task tabular regression benchmark, achieving an improvement of 0.338 in the median $R^2$ score and the lowest standard deviation across 34 tasks from the OpenML-CTR23 benchmark. Furthermore, our model has five times fewer parameters than the best-performing baseline and surpasses pretrained large language model baselines -- even when initialized from randomized weights.