In the construction of the bundle associated to some (right, say) $G$-principal bundle $P\to B$ and (left, say) action $G$ on a space $F$, we construct a bundle with typical fiber $F$ as the orbit space $P\times F/G$ where $G$ acts on $P\times F$ by $g\cdot(x,f)=(xg^{-1},gf)$.
Q1. Topologically, is it really required for $P$ to be a right $G$-principal bundle, as opposed to a left? It appears to me that everything works out with $g\cdot(x,f)=(gx,gf)$. We could also simply define a left action in a compatible manner in the usual way: $gx\mathop{=}\limits^{\scriptstyle\scriptstyle{\mathrm{def}}}xg^{-1}$.
Q2. If it is not strictly necessary from a topological standpoint, what motivates this definition?
(Probably silly questions, I know!)