I'm not familiar with the mower but your clutch appears similar to a motorcycle clutch.
You've got four plates that are keyed to the clutch basket. Between them and probably on the ends are plates that connect to the center shaft. The ears in the basket slots probably won't move enough closer together for you to see them move. Very little movement is needed to engage/disengage the clutch.
On a motorcycle there are springs that press the stack together and the clutch lever pushes a rod to unload the springs - disengaging the clutch. I wouldn't be surprised on the mower if the lever presses the plates together - you have to hold the lever to engage the clutch.
If the problem is that the clutch doesn't disengage I think boyscout862 nailed it for you - the plates are stuck together by corrosion. If the problem is it won't engage it's probably rust binding the plates with tabs to the basket and not letting them move. Either way, you'll need to clean up the rust.