In this work, we investigate the reasons that make symmetric partial verification essentially useless in virtually all domains. Departing from previous work, we consider any possible (finite or infinite) domain and general symmetric verification. We identify a natural property, namely, that the correspondence graph of a symmetric verification M is strongly connected by finite paths along which the preferences are consistent with the preferences at the endpoints, and prove that this property is sufficient for the equivalence of truthfulness and M-truthfulness. In fact, defining appropriate versions of this property, we obtain this result for deterministic and randomized mechanisms with and without money. Moreover, we show that a slightly relaxed version of this property is also necessary for the equivalence of truthfulness and M-truthfulness. Our conditions provide a generic and convenient way of checking whether truthful implementation can take advantage of any symmetric verification scheme in any (finite or infinite) domain. Since the simplest symmetric verification is the local verification, specific cases of our result are closely related, in the case without money, to the research about the equivalence of local truthfulness and global truthfulness. To complete the picture, we consider asymmetric verification and prove that a social choice function is M-truthfully implementable by some asymmetric verification M if and only if f does not admit a cycle of profitable deviations.