Suppose we have some group $G$, presented with some set of equations. If we are to collect up some subset of equations which define the presentation as $P$ (don't need all), then does swap of variables which preserve the actual group implied by the presentation $P$, tell us anything interesting about the group $G$?
Example:
Let's say we have for instance a group generated by $\{a,b,c\}$ with the relations $abc=b$, $abc =a$. Then in the for this group, we see that if we swap $a \leftrightarrow b$ , then the actual relation is same. The swap of variables give same equation hence the group implied is same. By implied group I mean, the group one gets when one applies the equations on the free group generated by the set.
By the way, It could also have been the case that we get different poly equation which imply the group as same.