An understanding of piano chord inversions and their implementation is most conducive to your having a more well-rounded cocktail piano style. The musical uses of inversions are many and placing a focus on them is a must for any aspiring cocktail pianist.
Being able to play any given chord in any position with both hands is a realistic goal worth setting for yourself if you haven't already. This will add variety to your left hand accompaniments for sure.
C E G spells out a C Major chord in root position. It's in root position because the name of the chord is at the bottom (furthest to the left on the keyboard) or "root" of the chord.
E G C also spells out the same chord. However, the chord has been inverted or "flipped." What was on the bottom of the chord in root position is now at the top. We call this C Major in 1st Inversion.
G C E is again the same chord yet inverted one more time, resulting in a C Major chord in 2nd inversion.
In addition, this opens up a whole new way of adding dimension to your piano playing when you can play any position of a chord on command with your right hand. You see, when you can play both the melody and the chord of the moment with your right hand, your left hand is free to take on a different role. You can actually, to a certain extent, "simulate" a three piece band. You can think of your melody as the singer or instrumentalist... the harmony as your "strumming" guitar player... and while this is going on, your left hand can actually take on the role of a bass player.
I feel strongly that this is an approach to piano playing that any aspiring stylist should explore. As a matter of fact, I considered it to be of enough importance to warrant my creating Right Hand Chord Piano Made Easy which shows how to actually incorporate this style. It can be used by anyone, including beginners, more advanced, and anyone in between. Once you understand the concept and apply it to some degree, you can take it to different levels, depending on your chord familiarity.
Even if you only know triads (three-note chords), you can take advantage of this program. As a matter of fact, I only use three basic chords (C, F, and G Major) to illustrate the use of this playing style throughout the tutorial. We take a look at piano chord inversions and their use can really add diversity to one's playing style.