Is there a name for an inverted state machine? I recently needed something like a state machine, but with a slightly different use case.

In general, I would say a state machine knows about a set of states, and different events. Depending on the current state and the event, a state machine is able to decide if this is a valid event for this state, and to which state it should transition to.

What I need is slightly different: Again, I have a state machine knowing some states, transitions and events, but it does tell me which events have to happen to transition to a certain state.
I already implemented such a machine and posted it for code review.

So, while the first machine is able to tell in which state it is after an event, the second machine is able to tell what has to happen to transition to a state.

According to this, my transition table looks different too.
Instead of having new states in the intersection of current states and events, I'm storing the events in the intersection of current and new states.

Is there a name for this kind of inverted state machine, or is it just another implementation of a finite 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.