Agitator doesn't work in wash and rinse cycles. It does spin to dump water after wash and spin cycles

I don't have much info on your washer like whether it is a front-loader or top-loader, etc. You need to check the motor coupler because this seems to be your problem.

Top load

This leads me to believe that the transmission is bad. But, is your washer the Centennial, the Bravos, or Epic washer? Here is a link to the Epic washer: https://dl.dropbox.com/u/8010163/8178604%20Maytag%20Epic%20Washer.pdf