@johncarlosbaez - Good point. We can define a right-(G)-action from a left (G)-action via[p\cdot g:=g^{-1}\cdot p] for all (p\in M,~g\in G). Then we can define a (G)-invariant function (or more generally a (G)-invariant tensor field) via[F(p):=\int_Gf(g^{-1}\cdot p)\mu_g.]for all (p\in M). If (\mu) is a left-invariant volume form on (G), then the proof that (F) is (G)-invariant now relies on the fact that (\mu) is left-invariant.
To me, this seems more natural. However, I don't think I've seen a paper or a book where a (G)-invariant structure is defined on a left (G)-space by inducing a right (G)-action. Everyone seems to start with a left (G)-action and a left Haar measure (or left-invariant volume form) on (G) and then defines the (G)-invariant structure using the original left (G)-action. Their proofs of (G)-invariance are always along the lines of ``... and since (\mu) is left-invariant it follows that"[F(x\cdot p)=\int_Gf(gx\cdot p)\mu_g=\int_Gf(g\cdot p)\mu_g=F(p).] even though the proof of this fact relies on (\mu) being right-invariant (which is true for (G) compact.) Not to pick on Hall's Lie group book but page 121 of Hall's book is an example.