My idea was to rotate it counterclockwise 45 degrees, putting the spigot about an inch further forward and to the right.
On first inspection, I wasn't able to figure out how to loosen the faucet, then I RTFM online and it was quite easy.
After consulting the manual again, it turns out the part limiting the rotation can be removed.
My brother helped with this project too, and we both struggled to get the target piece out. After messing with it for a while, I soaked it in CLR in case it too was calcified in place.
After getting everything back together, we have a rotated faucet with a freely rotating spigot. The change is subtle, but the functionality is definitely improved.