The Curiosity rover in Gale crater is investigating a mineral transition observed from orbit—an older "clay unit" to a younger "sulfate unit"—hypothesized to reflect the aridification of Mars' climate. Below this transition, the rover detected crystalline Ca-sulfates with minor Fe-sulfates but also found that some fraction of a rock's bulk SO₃ is often in the poorly constrained X-ray amorphous component. Here, we characterize the abundances and compositions of the X-ray amorphous sulfur-bearing phases in 19 drilled samples using a mass balance approach, and in a subset of 5 samples using evolved SO₂ gas measured using the SAM instrument. We find that ∼20-90 wt% of a sample's bulk SO₃ is in the X-ray amorphous state and that X-ray amorphous sulfur-bearing phase compositions are consistent with mixtures of Mg-S, Fe-S, and possibly Ca-S phases, likely sulfates or sulfites. These phases reside in the bedrock, perhaps as cementing agents deposited with detrital sediments or during early diagenesis, and in diagenetic alteration halos deposited after lithification during late diagenesis. The likely presence of highly soluble Mg-sulfates in the rocks suggests negligible fluid flow through the bedrock post-Mg-sulfate deposition. The X-ray amorphous sulfur-bearing phases probably became amorphous through dehydration in the current Martian atmosphere or inside the CheMin instrument. X-ray amorphous sulfur-bearing materials likely contribute to orbital spectral detections of sulfates, and so our results help form multiple hypotheses to be tested in the sulfate unit and are important for understanding the evolution of the Martian surface environment at Gale crater.