Sometimes I'll play different inversions. Like for example to do like a cm9 chord I might play a cm7 (C,Eb,G,Bb) with the left hand and a Ebmaj7 (Eb,G,Bb,D) with my right.... but I don't know.
not good to think in 'chords' with both hands. think about building chords using the whole keyboard, and throw the 'stacked thirds' theory of chords out the window, and just start from scratch. find the notes you need, on the whole keyboard, and use only the notes you need. it might only be one note, it might be 2. and use whatever hands you need, in order to play the notes you need, wherever they may be. Another way of looking at it -- if you are by yourself, play chords with your right hand and play bass (1 note!) with your left. Or, if you are playing the melody and chords, and someone else is the bass player, then i guess play chords in the middle range with your LEFT hand, and do the melody with your right hand. and if your band already has a bass player and someone else playing the melody, then tie one hand behind your back.