Rice's theorem for non-semantic properties
A general theorem that partially covers the example given is that any $\Sigma^0_1$-hard property of the machine will be undecidable. The halting problem is $m$-reducible to the state-reachability problem, so that shows the state reducibility problem is $\Sigma^0_1$-hard. 

However, this is not an "if and only if" theorem like Rice's theorem. If every $\Sigma^0_1$ property of the index of the Turing machine counts as a property of the machine, then there is not going to be a nice characterization, because there is not any nice characterization of which r.e. sets are decidable in terms of the index of the r.e. set.