Say I want to put action of $S_4$ on $V^{\otimes 4}$. If I want to act on the left, why can't I say
$\sigma(v_1 \otimes v_2 \otimes v_3 \otimes v_4) = v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(4)}?$
Everything seems to work out right e.g. take $\sigma = (123),\tau = (134)$. Then $\sigma\tau = 234$.
$(\sigma\tau)(v_1 \otimes \ldots \otimes v_4) = v_1 \otimes v_3 \otimes v_4 \otimes v_2$
and $\sigma\big( \tau(v_1 \otimes \ldots \otimes v_4) \big)$ is also the same thing. But books I see say we have to throw in an inverse for the left action? I am getting confused because it seems throwing the inverse does not make it an action! What's happening?