What do I do when my analysis gets twisted to say what people want?

You do careful, honest work, and then you watch it bent to fit a conclusion someone already wanted. That is a particular kind of pain, because your integrity is the whole point of what you do. Feeling the weight of that compromise means your conscience is still working. Do not lose that. You control the analysis. You do not always control what others do with it once it leaves your hands. Hold that line clearly in your own mind, so their misuse does not become your guilt. The distortion is theirs, not yours. Still, do what you can. Write your caveats plainly, put your real findings on record, say clearly what the data does and does not support. Where the twisting is deliberate and constant, decide how long you can stand in a place that asks you to bend. Your name and your peace are worth protecting.