Update: here's a good solution I got from an Esri friend. For reference, my raster values went from -1 to 0.796.
In the Primary Symbology Pane for your raster, you’ll need to go to the Statistics panel towards the bottom. There is a dropdown menu titled “statistics”. Change it from “Dataset” to “Custom”. Look at your min and max values for whichever has the greatest absolute value, and then set min to the negative of that number and max to the positive of that number. In your case you’ll want Min to be -1 and Max to be 1. This will ensure that:
- zero is the center of your color scheme
- your min or your max is at one end of the color scheme (in your case, the minimum is the left end)
- the remaining point is somewhere between the center and the remaining end (in your case, the max is between the middle and the right end)
If you want to make sure that your desired color is used for your maximum, you can open the dropdown for the color scheme and click “Format Color Scheme”. You conveniently have values between -1 and 1, so it’s relatively simple to get the maximum value’s position in the color scheme. It will be 1 - ((1 - 0.796) / 2) = 89.8%, so you can select the leftmost color stop on the gradient and then type this value into the Position field. The stops for -1 and 0 should remain at 0% and 50%.