Is there a name for an inverted state machine?
If I understand you correctly, you could achieve this using transducers (i.e. automata which also produce outputs when taking a transition).

Suppose we are given a DFA $A = (Q,q_0,\Sigma,\delta)$ describing which events lead to which state transitions. Here $Q$ is the set of states, $q_0$ is the initial state, $\Sigma$ the alphabet of events, and $\delta:Q\times\Sigma\to Q$ the transition function. Then we can define a (nondeterministic) transducer $A'$ as $(Q,q_0,Q,\Sigma,\delta')$ as follows:


$Q$ is the same set of states as before, and $q_0$ is still initial;
The input alphabet is now also $Q$ (representing the target state you are trying to reach);
$\Sigma$ is now the output alphabet;
The transition relation $\delta'$ contains a tuple $(q,\delta(q,a),a,\delta(q,a))$ for each $q\in Q$ and $a\in\Sigma$ (representing the fact that to get from $q$ to $r$, you can use $a$ if $r=\delta(q,a)$). 


Note that I'm assuming that you actually want to proceed to the given target state, otherwise you need to adapt the transition relation. Also, if there is no transition from $q$ to $r$, the computation simply fails; you may want to handle this case differently.