Hadamard preprocessing is widely used in large language model (LLM) quantization to reduce activation coherence and suppress peaks. Existing methods typically rely on random sign flips before the Hadamard transform, which do not use information from the target data distribution. We propose Greedy Hadamard Alignment (GHA), a simple training-free method to build a signed-permutation preconditioner from calibration data. GHA uses a pooled representative activation, a greedy search over flat targets, and an efficient alignment step that is analytically optimal once the target is fixed. We also study when one shared GHA preconditioner can transfer across LLM activations. GHA works best when activations are directionally aligned and stable across calibration sources. Overall, GHA improves post-quantization model quality over randomized Hadamard baselines, with the strongest gains in KV-cache quantization, more moderate gains for hidden states, and little compute overhead during inference.