You may have a worn agitator cam assembly, worn agitator dogs, or a loose agitator bolt.
Whirlpool uses a split agitator design. The upper portion is called the auger, while the bottom is called the agitator. Between these components is a agitator cam assembly. This problem is commonly caused by either worn agitator dogs, a worn agitator cam assembly, or a loose agitator bolt. The following link explains what the dogs are and how to access and/or replace them. It also explains how to access the bolt and tighten (if necessary):http://www.fixya.com/support/r3608577-replacing_agitator_dogs
The part number for the dogs are included in the link provided. You can purchase replacements at pcappliancerepair.com for about $5. For a little more you can purchase the entire cam replacement kit, which includes the cam, the thrust bearing, and the dogs for about $11 or $12.
NOTE: You can refer to the excellent exploded view parts diagrams the web site has to offer for assistance. Whirlpool also recommends the application of a little grease on the thrust washer. The part number for their recommended grease is AP3098553.
If you aren't sure, or have additional questions about this repair, please post back with your complete model number (located on a nameplate along the wash tub opening under the lid) and let me know. I hope this helps you.
NOTE: DO NOT overtighten the agitator bolt. This can cause damage to the plastic, or restrict the agitator movement. Tighten until snug only.