Jitter-type spike resampling methods are routinely applied in neurophysiology for detecting temporal structure in spike trains (point processes). Several variations have been proposed. The concern has been raised, based on numerical experiments involving Poisson spike processes, that such procedures can be conservative. We study the issue and find it can be resolved by reemphasizing the distinction between spike-centered (basic) jitter and interval jitter. Focusing on spiking processes with no temporal structure, interval jitter generates an exact hypothesis test, guaranteeing valid conclusions. In contrast, such a guarantee is not available for spike-centered jitter. We construct explicit examples in which spike-centered jitter hallucinates temporal structure, in the sense of exaggerated false-positive rates. Finally, we illustrate numerically that Poisson approximations to jitter computations, while computationally efficient, can also result in inaccurate hypothesis tests. We highlight the value of classical statistical frameworks for guiding the design and interpretation of spike resampling methods.