Showerhead was surprisingly ugly beforehand (oh, the things we don't notice!). Soaked it in white vinegar with the showerhead down, in a bag in the sink. Left it about an hour or so. Scrubbed off the remainder with an old toothbrush.
showerhead-01.jpg
Before
showerhead-02.jpg
Trying to get it attached on the shower; couldn't get it right
showerhead-03.jpg
Took the showerhead off and soaking in the bag of vinegar in the sink