Block swapping can mimic time irreversibility

Swapping two blocks of observations changes which block comes first, but keeps the observations inside each block in the same order. Reversing the whole record changes both orders. We study this difference in models with a finite number of possible states. The next-state probability depends only on the current state. The probability of an observation pattern does not change when we shift its position in time, and every transition has a probability greater than zero. In one two-state model, every record is as likely as its full reverse. Yet the block-swap comparison gives a positive value. We derive an exact formula for this comparison. When each block contains at least two observations, the result is zero exactly when successive observations are independent. As block length tends to infinity, the result converges to a measure of dependence between neighbouring observations. These findings provide exact model checks for calculations intended to measure time-reversal differences. We do not reanalyse experimental neural recordings.