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We present a novel bi-directional Transformer architecture (BiXT) which scales linearly with input size in terms of computational cost and memory consumption, but does not suffer the drop in performance or limitation to only one input modality seen with other efficient Transformer-based approaches. BiXT is inspired by the Perceiver architectures but replaces iterative attention with an efficient bi-directional cross-attention module in which input tokens and latent variables attend to each other simultaneously, leveraging a naturally emerging attention-symmetry between the two. This approach unlocks a key bottleneck experienced by Perceiver-like architectures and enables the processing and interpretation of both semantics (`what') and location (`where') to develop alongside each other over multiple layers -- allowing its direct application to dense and instance-based tasks alike. By combining efficiency with the generality and performance of a full Transformer architecture, BiXT can process longer sequences like point clouds or images at higher feature resolutions and achieves competitive performance across a range of tasks like point cloud part segmentation, semantic image segmentation and image classification.