IN addition to what has been said, I would like to mention the following aspect: a group action usually does not come alone. When $G$ acts on a set $X$ you will almost for sure be interested also in the induced action of $G$ on certain kind of functions on $X$: think of a smooth manifold $M$ and of smooth functions, or of smooth tensor fields on $M$, or what not. Then the induced group action can be either viewed as a right action when $G$ acts from the left on $X$ or you have to plug in a $^{-1}$ to turn things again into a left action. In various situations (say equivariant maps in the dual of a Lie algebra etc) it might become a notational desaster if you want to specify all kind of actions with a separate symbol. However, it might be important to keep track of whether you have a left or a right action.

So my habit is to denote the action of a group element $g$ on some object $x \in X$ by $g \triangleright x$ if it is a left action and by $g \triangleleft x$ or better $x \triangleleft g$ if it is a right action. Then you don't have to bother whether you have already included the $^{-1}$ in the coadjoint action to make it a left action or just stay with a right action :)