I have been to Chaska, MN - nice small town with lots of creamy yellow brick buildings.... Chaska brick weighed less than other brick, making it cheaper to transport....

http://www.chaskabrick.com/

"Many important buildings in Minnesota are constructed of Chaska brick.... After production ended, its historic significance was
recognized. Many buildings made of Chaska brick were placed on the National
Register of Historic Places."

Please don't paint it... :)

http://www.mnopedia.org/group/chaska-brick-industry-1857-1950

Thank you (v. belatedly!) for your comment. Am still trying to determine what this brick is. I've looked at the Chaska brick and it doesn't look like a fit. Ours is very rough--almost like a pumice/volcanic texture. It was likely installed in our (1922) house in a 1950's renovation and doens't appear to have been painted. It appears that it might have some matte glaze on it.

What we really want is to clean it, so (after trying a few things myself) I think I am stuck calling a masonry expert.

Any other comments are appreciated!

This might be an obvious question, but have you tried power washing it? Perhaps with bleach?