Have an axe to grind
There were once people who travelled around with grindstones, large stones used to sharpen knives and other cutting blades such as axes. The grindstone was often circular in shape and turned like a wheel while the blade was held against it. Moving the heavy grindstone was hard work.
When you have your nose to the grindstone, you are working very hard. It is probably why some people liked to get others to grind or sharpen their axes.
American founding father Benjamin Franklin had a story about a clever man who would come over and start praising Franklin's grindstone and the way it worked. The man passed him his axe and asked Franklin to demonstrate. Franklin soon realised he had been tricked into helping the man. Writer Charles Miner turned the anecdote into a story about a man flattering a boy into sharpening his axe for him and then scolding him for being late to school.
Someone who has an axe to grind has a hidden motive and is not as neutral as he seems. Nowadays it is not used very much to describe people who are trying to trick you. It's more likely to refer to someone with a hidden motive or agenda. A reviewer may say a book is poor and disorganised, but you then discover that the author took away the reviewer's girlfriend. The reviewer had an axe to grind.
Lose your rag