I figured out what was causing it, but I'm still fuzzy on the exact reason why. In Photoshop, you can choose between various rendering intents. When I use "Relative Colormetric" it prints white as no ink. If I choose "Absolute Colormetric" it print white as a very light grey.
From the descriptions of these in Photoshop and other places, I don't really understand why Absolute Colormetric would have this problem. Here is Photoshop's description of Absolute Colormetric:
Colors that fall inside the destination gamut are unchanged. Out-of-gamut colors are shifted to the closest reproducible color. No scaling of colors to destination white point is performed. This intent maintains color accuracy at the expense of preserving relationships between colors.
My best guess is that for whatever reason, Photoshop decides that white is out-of-gamut and shift it to the very lightest gray that the printer can produce. It seems to me that white should always be in-gamut, but I'm new to all this.