Let's say I cast Freedom of Movement on myself. The next round I initiate grapple, can I initiate a grapple? and if successful would that result in the opponent being grappled and myself free of the grappling penalty (which normally one would suffer a -20 penalty to try to succeed).
Freedom of movement does not make your character immune to the grappled condition. Meaning that if you initiated the grapple, you still gain the grappled condition.
All combat maneuver checks made to grapple the target automatically fail.
But any attempts to revert the grapple will automatically fail.