For a group $G$ there is a natural notion of left action on subsets of $G$ given by $g \triangleright H := gH$. But simultaneously, there is a natural right action as well: $H \triangleleft g := Hg$. And furthermore, they seem to satisfy a commutativity relation: $(g \triangleright H) \triangleleft h = g \triangleright (H \triangleleft h)$.
The situation seems identical to bimodules in ring theory.
I'm wondering if these are studied anywhere. They seem like they would give a slight shift in perspective when doing the usual subset manipulations in basic group theory.
If these things have been studied before, is there a common name for them? (The Google results for "group biaction" did not serve me well...). What other examples might there be where a structure has two natural compatible actions?
You can talk about left, right, and two-sided actions of monoids, and in particular of groups. An $(M, N)$ biaction, where $M, N$ are two monoids, is the same thing as an $M \times N^{op}$ action, where $N^{op}$ is the opposite monoid.
A typical example is that in any category, if $c, d$ are two objects, then $\text{Hom}(c, d)$ has an $(\text{End}(d), \text{End}(c))$ biaction.